A DOA estimation method based on geometric distance metric under low signal-to-noise ratio

Abstract

The present invention provides a DOA estimation method based on geometric distance metric under low signal-to-noise ratio. The method constructs a signal model of a sensor array according to the direction vector of the signal source, introduces information geometric metric, gives a definition of geometric distance metric, selects a reference matrix for implementing DOA estimation through geometric distance metric, performs regularization processing on the selected reference matrix, solves the distance of geometric distance metric, and obtains the estimated target direction. The present invention transforms the direction estimation problem into a study of the geometric structure difference problem between two points on a matrix manifold, and can well implement DOA estimation through geometric structure difference. After selecting the reference matrix and performing reversible processing on the reference matrix, it is substituted into the information geometric distance calculation formula of LEM, KLD, SKLD and JBLD to obtain the geometric distance values corresponding to different angles. The angle corresponding to the minimum geometric distance value is the target direction. Under the conditions of low signal-to-noise ratio and small number of snapshots, the target direction estimation has a high estimation accuracy.

Description

Technical Field

[0001] The present invention relates to the field of signal processing technology, and in particular to a DOA estimation method under low signal-to-noise ratio. Background Art

[0002] Estimating the direction of arrival (DOA) of array signals is a crucial technical issue in array signal processing, which is widely used in fields such as radar, communications, electronics, and biomedicine. A major research area within array signal processing is spatial spectrum estimation, which is widely used in signal source position estimation.

[0003] Initial DOA estimation methods were based on traditional beamforming algorithms using array signal processing. However, the Rayleigh limitation of arrays limits the spatial resolution of these algorithms. Consequently, various high-resolution DOA estimation methods have emerged. These methods theoretically overcome the Rayleigh criterion for azimuth resolution and can be categorized into three main approaches. The first is the subspace-based spatial spectrum estimation algorithm based on eigendecomposition. Based on different approaches to processing the eigenspace, subspace-based algorithms can be divided into two categories: noise subspace-based algorithms, exemplified by the Multiple Signal Classification (MUSIC) algorithm, and signal subspace-based algorithms, defined by the Estimation of Signal Parameters through Rotational Invariance Technique (ESPRIT) algorithm. Currently, several super-resolution DOA estimation methods are based on the principles of these two subspace-based algorithms, such as the ROOT-MUSIC algorithm, the Multidimensional MUSIC algorithm, the LS-ESPRIT algorithm, the TLS-ESPRIT algorithm, and the Conjugate ESPRIT algorithm. The second approach, DOA estimation based on subspace fitting, applies maximum likelihood parameter estimation methods to DOA estimation. Representative methods include the maximum likelihood (ML) algorithm, the weighted subspace fitting (WSF) algorithm, and the multidimensional MUSIC algorithm. The third approach, DOA estimation based on compressed sensing, is used when the signal is sparse or compressible. However, these methods can suffer from large estimation errors under low signal-to-noise ratio (SNR) and small snapshot support conditions. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies, the present invention provides a DOA estimation method based on a geometric distance metric under low signal-to-noise ratio (SNR) conditions. This method proposes DOA estimation within the framework of matrix information geometry theory, transforming the DOA estimation problem into a problem of geometric structural differences between two points on a matrix manifold. For two points on a manifold, a distance metric is used to measure the distance between them. A larger distance indicates a greater local geometric structural difference between the two points, and thus greater discrimination. Consequently, the minimum distance measured by the geometric distance metric indicates the target's direction.

[0005] The technical solution adopted by the present invention to solve the technical problem includes the following steps:

[0006] The first step is to construct the signal model X(t) of the sensor array based on the direction vector of the signal source;

[0007] The second step is to introduce information geometry metrics and give the definition of geometric distance metrics;

[0008] The third step is to select the reference matrix for DOA estimation using geometric distance metric;

[0009] The fourth step is to regularize the selected reference matrix;

[0010] The fifth step is to solve the distance measured by geometric distance to obtain the estimated target position.

[0011] In the first step, K far-field narrowband target signal sources are incident on a uniform linear array with N elements. The element spacing of the uniform linear array is d, where λ is the wavelength corresponding to the system operating frequency f0, λ = c / f0, c is the speed of signal propagation in the medium, and the received additive noise is independent of each other and is a stationary, zero-mean Gaussian white noise with a variance of δ 2 , it can be seen that the signal model of the N-element array receiving signal is expressed as:

[0012] X(t)=AS(t)+N(t)

[0013] Where X(t) is the signal model of the array, S(t) is the target signal vector, N(t) is the array additive noise vector, L is the number of sampling snapshots of the array receiving the signal vector, A is the steering vector matrix, and:

[0014]

[0015] Where,

[0016] Where k = 1, 2, ..., K, α k is the incident angle of the kth signal;

[0017] The covariance matrix of the received signal is obtained as follows:

[0018]

[0019] In the formula and Represents the covariance matrix R x The signal and noise components, and denote the signal power and noise power respectively, (·) Hdenotes the conjugate transpose, diag(·) denotes a diagonal matrix, and I denotes the identity matrix.

[0020] The second step introduces matrix information geometry and gives the definition of four geometric distance metrics; the geometric distance on the manifold reflects the geometric structure of the manifold, and different geometric distances reflect different geometric structures and have different abilities to distinguish two points on the manifold. On the matrix manifold, in addition to defining geodesic distances, such as the Log-Euclidean Metric (LEM), many divergence metrics are also defined, such as the Kullback-Leibler Divergence (KLD), the Symmetric Kullback-Leibler Divergence (SKLD), and the Jensen-Breg-man LogDet Divergence (JBLD). These divergences all have some good properties and have been widely used. The specific definitions of the four geometric distance metrics are given below;

[0021] For two points X1 and X2 on the matrix manifold, the LEM distance between them is:

[0022] d L (X1,X2)=||log(X1)-log(X2)|| F

[0023] For two points X1 and X2 on the matrix manifold, the KLD distance between them is:

[0024]

[0025] For two points X1 and X2 on the matrix manifold, the SKLD distance between them is:

[0026]

[0027] For two points X1 and X2 on the matrix manifold, the JBLD distance between them is:

[0028]

[0029] In the formula, ||·|| F Indicates taking the F norm, tr(·) indicates the trace of the matrix, I indicates the identity matrix, log(·) indicates taking the logarithm, and |·| indicates taking the determinant value.

[0030] The third step involves selecting a reference matrix. Since DOA estimation is based on matrix information geometry, the DOA estimation problem is transformed into studying the geometric structure differences between two points on the matrix manifold. The smaller the difference, the more similar the two points on the rectangle are, and the smaller the geometric distance is. The angle at which the geometric distance is minimized is the target's incident direction. At the same time, when receiving the target echo signal, noise and various interferences are inevitably encountered. Due to the high randomness of noise, in order to reduce its impact on estimation accuracy, when selecting a reference matrix, it is necessary to avoid matrices containing noise as much as possible. Therefore, selecting an appropriate reference matrix is very important.

[0031] The reference matrix selected in the fourth step is:

[0032] R1=E[(a(θ i )·S(t))X H ]

[0033] R2=E[(a(θ i )·S(t))(a(θ i )·S(t)) H ]

[0034] In the formula, E(·) represents the expectation, θ i ∈(-90°,90°) represents the scanning angle, a(θ i ) represents the array manifold corresponding to the scanning angle.

[0035] The fourth step is to solve the problem that the selected reference matrix is irreversible, and the reference matrices R1 and R2 need to be converted into reversible matrices. First, the conjugate transpose matrix of the reference matrix R1 and the reference matrix R2 is converted into Add together to get a symmetrical reference matrix The transformation equation is as follows:

[0036]

[0037] In order to ensure the reversibility of the reference matrices R1 and R2, a loading factor D = εI is added, where ε is an infinitesimal constant and I is the identity matrix, thus obtaining two symmetric positive definite reference matrices and The formula is as follows:

[0038]

[0039]

[0040] The fifth step solves the distance measured by the geometric distance to obtain the estimated target direction. The basic principle of matrix information geometric DOA estimation is to transform the direction estimation problem into a geometric problem on the matrix manifold for research, and to use the difference in the geometric structure of the matrix manifold to distinguish the characteristic information of the target signal and the clutter. Therefore, a reference matrix containing the target signal is selected on the matrix manifold. The geometric distance between the two reference matrices is compared to estimate the target direction. The estimation principle is given by the following formula

[0041]

[0042] Where θ tag represents the target direction, and d(·) represents the geometric distance between the two matrices. Thus, it can be seen that since the geometric distances corresponding to different angles within the scanning angle are different, the angle corresponding to the minimum geometric distance is the direction of the target, thus achieving DOA estimation.

[0043] The beneficial effect of the present invention is that DOA estimation is proposed within the framework of matrix information geometry theory, and the problem of azimuth estimation is transformed into the study of the geometric structure difference between two points on the matrix manifold. DOA estimation can be well achieved through the geometric structure difference. After selecting a suitable reference matrix and performing reversible processing on the reference matrix, it is substituted into the information geometry distance calculation formula of LEM, KLD, SKLD and JBLD to obtain the geometric distance values corresponding to different angles, where the angle corresponding to the minimum geometric distance value is the target azimuth. This method still has a high estimation accuracy for target azimuth estimation under the conditions of low signal-to-noise ratio and small number of snapshots. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 It is a structural diagram of a uniform linear array of N elements.

[0045] Figure 2 is the estimated mean square error of the LEM, KLD, SKLD and JBLD methods under different signal-to-noise ratios. DETAILED DESCRIPTION

[0046] The present invention will be further described below with reference to the accompanying drawings and examples.

[0047] In the first step of the embodiment, it is assumed that there are K far-field narrowband target signal sources incident on a uniform linear array with N=8 elements, and the element spacing of the uniform linear array is d=λ2, where λ is the wavelength corresponding to the system operating frequency f0, λ=c / f0, c is the speed of signal propagation in the medium, and the received additive noise is independent of each other and is a stationary, zero-mean Gaussian white noise with a variance of δ 2 , it can be seen that the mathematical model of the N-element array receiving signal is expressed as:

[0048] X(t)=AS(t)+N(t)

[0049] Where X(t) is the array snapshot data vector, S(t) is the target signal vector, N(t) is the array additive noise vector, L = 1024 is the number of sampled snapshots of the array receiving signal vector, A(θ) is the N × K dimensional steering vector matrix, and:

[0050]

[0051] Where,

[0052] Where k = 1, 2, ..., K, α k is the incident angle of the kth signal;

[0053] The covariance matrix of the received signal is obtained as follows:

[0054]

[0055] In the formula and Represents the covariance matrix R x The signal and noise components, and denote the signal power and noise power respectively, (·) H denotes the conjugate transpose, diag(·) denotes a diagonal matrix, and I denotes the identity matrix.

[0056] The second step introduces matrix information geometry and gives the definition formulas of four geometric distance metrics. The geometric distance on the manifold reflects the geometric structure of the manifold. Different geometric distances reflect different geometric structures and have different distinguishing abilities between two points on the manifold. On the matrix manifold, in addition to defining geodesic distances, such as the Log-Euclidean Metric (LEM), many divergence metrics are also defined, such as the Kullback-Leibler Divergence (KLD), the Symmetric Kullback-Leibler Divergence (SKLD), and the Jensen-Breg-man LogDet Divergence (JBLD). These divergences all have some good properties and have been widely used. The specific definitions of the four geometric distance metrics are given below.

[0057] For two points R1 and R2 on the matrix manifold, the LEM distance between them is:

[0058] d L(R1,R2)=||log(R1)-log(R2)|| F

[0059] For two points R1 and R2 on the matrix manifold, the KLD distance between them is:

[0060]

[0061] For two points R1 and R2 on the matrix manifold, the SKLD distance between them is:

[0062]

[0063] For two points R1 and R2 on the matrix manifold, the JBLD distance between them is:

[0064]

[0065] In the formula, ||·|| F Indicates taking the F norm, tr(·) indicates the trace of the matrix, I indicates the identity matrix, log(·) indicates taking the logarithm, and |·| indicates taking the determinant value.

[0066] The third step is to select a suitable reference matrix. Since DOA estimation is based on matrix information geometry, the DOA estimation problem is transformed into studying the difference in geometric structure between two points on the matrix manifold. The smaller the difference, the more similar the two points on the rectangle are, and the smaller the geometric distance is. The angle at which the geometric distance is minimum is the incident direction of the target. At the same time, in the process of receiving the target echo signal, noise and various interferences will inevitably be encountered. Since the randomness of noise is relatively large, in order to reduce the impact of noise on estimation accuracy, when selecting a reference matrix, it is necessary to try to avoid matrices containing noise. Therefore, it is very important to select a suitable reference matrix.

[0067] The reference matrix selected by the present invention is:

[0068] R1=E[(a(θ i )·S(t))X H ]

[0069] R2=E[(a(θ i )·S(t))(a(θ i )·S(t)) H ]

[0070] In the formula, E(·) represents the expectation, θ i ∈(-90°,90°) represents the scanning angle, and is discretely valued at equal intervals of Δθ=0.1°. a(θ i ) is represented as the array manifold corresponding to the scanning angle.

[0071] The fourth step is to solve the problem that the selected reference matrix is irreversible, and the reference matrices R1 and R2 need to be converted into reversible matrices. First, the conjugate transpose matrix of the reference matrix R1 and the reference matrix R2 is converted into Add together to get a symmetrical reference matrix The transformation equation is as follows:

[0072]

[0073] In order to ensure the reversibility of the reference matrices R1 and R2, a loading factor D = εI is added, where ε is an infinitesimal constant and I is the identity matrix, thus obtaining two symmetric positive definite reference matrices and The formula is as follows:

[0074]

[0075]

[0076] The fifth step solves the distance measured by the geometric distance to obtain the estimated target direction. The basic principle of matrix information geometric DOA estimation is to transform the direction estimation problem into a geometric problem on the matrix manifold for research, and to use the difference in the geometric structure of the matrix manifold to distinguish the characteristic information of the target signal and the clutter. Therefore, a reference matrix containing the target signal is selected on the matrix manifold. The geometric distance between the two reference matrices is compared to estimate the target direction. The estimation principle is given by the following formula

[0077]

[0078] Where θ tag represents the target direction, d(·) represents the geometric distance between the two matrices, and R1 and R2 represent the selected reference matrices. Thus, it can be seen that since the geometric distances corresponding to different angles within the scanning angle are different, the angle corresponding to the minimum geometric distance is the direction of the target, thus achieving DOA estimation.

Claims

1. A DOA estimation method based on geometric distance metric under low signal-to-noise ratio, characterized by The steps include: The first step is to construct the signal model X(t) of the sensor array based on the direction vector of the signal source; The second step is to introduce information geometry metrics and give the definition of geometric distance metrics; The third step is to select the reference matrix for DOA estimation using geometric distance metric; The fourth step is to regularize the selected reference matrix; The reference matrix selected in the fourth step is: In the formula, E(·) represents the expectation, θ i ∈(-90°,90°) represents the scanning angle, a(θ i ) represents the array manifold corresponding to the scanning angle; Convert the reference matrices R1 and R2 into reversible matrices; first convert the reference matrix R1 to the conjugate transposed matrix of the reference matrix R1 Add together to get a symmetrical reference matrix The transformation equation is as follows: In order to ensure the reversibility of the reference matrices R1 and R2, a loading factor D = εI is added, where ε is an infinitesimal constant and I is the identity matrix, resulting in: Thus we get two symmetric positive definite reference matrices and The fifth step is to solve the distance measured by geometric distance to obtain the estimated target position.

2. The DOA estimation method based on geometric distance metric under low signal-to-noise ratio according to claim 1, characterized in that: In the first step, K far-field narrowband target signal sources are incident on a uniform linear array with N elements. The element spacing of the uniform linear array is d, where λ is the wavelength corresponding to the system operating frequency f0, λ = c / f0, c is the speed of signal propagation in the medium, and the received additive noise is independent of each other and is a stationary, zero-mean Gaussian white noise with a variance of δ 2 , it can be seen that the signal model of the N-element array receiving signal is expressed as: X(t)=AS(t)+N(t) Where X(t) is the signal model of the array, S(t) is the target signal vector, N(t) is the array additive noise vector, L is the number of sampling snapshots of the array receiving the signal vector, A is the steering vector matrix, and: Where, Where k = 1, 2, ..., K, α k is the incident angle of the kth signal; The covariance matrix of the received signal is obtained as follows: In the formula and Represents the covariance matrix R x The signal and noise components, and denote the signal power and noise power respectively, (·) H denotes the conjugate transpose, diag(·) denotes a diagonal matrix, and I denotes the identity matrix.

3. The DOA estimation method based on geometric distance metric under low signal-to-noise ratio according to claim 1, characterized in that: The second step introduces information geometry metrics and gives the definitions of four geometric distance metrics; the specific definitions of the four geometric distance metrics are: For two points X1 and X2 on the matrix manifold, the LEM distance between them is: d L (X1,X2)=log(X1)-log(X2) F For two points X1 and X2 on the matrix manifold, the KLD distance between them is: For two points X1 and X2 on the matrix manifold, the SKLD distance between them is: For two points X1 and X2 on the matrix manifold, the JBLD distance between them is: Where, F Indicates taking the F norm, tr(·) indicates the trace of the matrix, I indicates the unit matrix, log(·) indicates taking the logarithm, and · indicates taking the determinant value.

4. The DOA estimation method based on geometric distance metric under low signal-to-noise ratio according to claim 1, characterized in that: The fifth step solves the distance measured by the geometric distance to obtain the estimated target orientation, selects a reference matrix containing the target signal on the matrix manifold, and compares the size of the geometric distance between the two reference matrices to estimate the target orientation. The estimation principle is given by the following formula: Where θ tag represents the target direction, and d(·) represents the geometric distance between the two matrices. Since the geometric distances corresponding to different angles within the scanning angle are different, the angle corresponding to the minimum geometric distance is the direction of the target, thereby realizing DOA estimation.

Images