Super-resolution direction-of-arrival estimation method for uniform linear array based on inverse beamforming

Abstract

The method for super-resolution direction of arrival estimation of uniform linear array based on inverse beamforming comprises the following steps: step (1): constructing a narrowband single snapshot echo signal model of the linear array, realizing the signal model of direction of arrival estimation, and forming a beam; step (2): constructing a discrete Fourier transform matrix F N Inverse beamforming observation model; Step (3): Construct the discrete Fourier transform matrix F P Inverse beamforming observation model; Step (4): Construct the downsampling matrix Θ, N actual array data y NP Considered as the full sampling data y of P virtual array elements using Θ P The result of downsampling; Step (5): Based on the inverse beamforming observation model at point P, use Θ, F P and y NP , constructing a P (P>N) point inverse beamforming super-resolution observation model; step (6): constructing a super-resolution optimization model based on the weighted L1 norm, using MATLAB's CVX toolbox to solve the optimization model, and finally achieving direction of arrival super-resolution estimation. The present invention achieves direction of arrival super-resolution estimation of linear arrays.

Description

Technical Field

[0001] The present invention belongs to the technical field of super-resolution direction of arrival estimation of linear arrays, and in particular relates to a super-resolution direction of arrival estimation method for a uniform linear array based on inverse beamforming. Background Art

[0002] Super-resolution direction-of-arrival (DOA) estimation for uniform linear arrays is an important research area in signal processing, particularly in radar, sonar, and wireless communication systems. Super-resolution technology can overcome the limitations of traditional Nyquist sampling theorem and provide the ability to resolve more signal sources than the number of array elements.

[0003] The MUSIC (Multiple Signal Classification) algorithm and the ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) algorithm are two classic super-resolution DOA estimation methods. The MUSIC algorithm decomposes the received signal's spatial spectrum into signal and noise subspaces and uses the orthogonality of the signal subspaces to estimate the signal's direction of arrival. The ESPRIT algorithm uses the translational invariance of the array to estimate the signal's direction of arrival. Although the MUSIC and ESPRIT algorithms have high resolution in DOA estimation, their performance is limited by the signal-to-noise ratio and the array's geometric structure.

[0004] In recent years, compressed sensing (CS) theory has been applied to DOA estimation. This method exploits the sparse nature of signals and solves a sparse representation of the signal through an optimization problem, thereby achieving super-resolution DOA estimation. CS-based DOA estimation methods are suitable for situations where the number of sources is unknown or far less than the number of array elements. They are often used to estimate the direction of arrival of signals in large arrays and in low signal-to-noise ratio environments.

[0005] In addition to the aforementioned methods, deep learning-based DOA estimation methods have gradually become a mainstream research topic. These methods, with their powerful feature extraction capabilities, good generalization, ability to handle complex situations, and support for end-to-end learning, are suitable for super-resolution DOA estimation in uniform linear arrays. However, these methods also have limitations, such as high resource consumption and the risk of overfitting. Therefore, when applying deep learning to DOA estimation, these advantages and disadvantages must be comprehensively considered, and appropriate adjustments and optimizations must be made for specific application scenarios.

[0006] There are currently three main types of mainstream algorithms. One is the traditional covariance algorithm, such as MUSIC; one is the covariance algorithm that utilizes the sparsity of DOA in the angle domain, such as the direction of arrival super-resolution estimation method based on compressed sensing; and the other is the covariance algorithm based on deep learning.

[0007] All of the above algorithms achieve good high-resolution DOA estimation. However, most of these methods are based on two key assumptions: the first is that the number of data snapshots is much larger than the number of antenna arrays, and the second is that the received signals are incoherent. These two assumptions produce a covariance matrix whose rank from DOA is equal to the number of targets. When either of these conditions is violated, the rank of the DOA decreases, rendering most multi-snapshot DOA estimation methods ineffective or suffering severe performance degradation. Summary of the Invention

[0008] To overcome the shortcomings of the aforementioned prior art, the present invention aims to provide a method for super-resolution direction of arrival estimation for uniform linear arrays based on inverse beamforming. This method, based on beamforming, constructs a super-resolution direction of arrival observation model based on discrete Fourier transform (DOT), ultimately achieving super-resolution DOT estimation for linear arrays. This observation model maintains the matrix dimension and array domain redundant information, making the related optimization problems easy to solve. Due to the use of a model construction method based on beamforming (DBF) and fast Fourier transform (FFT), the proposed super-resolution observation model is universally applicable and easy to implement.

[0009] In order to achieve the above object, the technical solution adopted by the present invention is:

[0010] The method for estimating direction of arrival of a uniform linear array super-resolution based on inverse beamforming comprises the following steps:

[0011] Step (1): Construct a narrowband single snapshot echo signal model of the linear array, implement the signal model for direction of arrival estimation, and form a beam;

[0012] Step (2): Based on beamforming, construct a discrete Fourier transform matrix F N The inverse beamforming observation model of

[0013] Step (3): Assume that there are P (P>N) array elements with full sampling data y P Based on the observation model, full sampling data expansion is performed to construct a discrete Fourier transform matrix F P The inverse beamforming observation model of

[0014] Step (4): Use virtual array element downsampling to construct the downsampling matrix Θ, N actual array element data y NPConsidered as the full sampling data y of P virtual array elements using Θ P The result of downsampling;

[0015] Step (5): Based on the inverse beamforming observation model at point P, use Θ, F P and y NP , construct a P (P>N) point inverse beamforming super-resolution observation model;

[0016] Step (6): Construct a super-resolution optimization model based on the weighted L1 norm, use MATLAB's CVX toolbox to solve the optimization model, and finally achieve super-resolution estimation of the direction of arrival.

[0017] The narrowband single snapshot echo signal model characteristics of the linear array constructed in step (1) are as follows:

[0018] Assume a linear array with N elements, and its narrowband single snapshot echo signal model is expressed as

[0019] y=Γs+e noise (1)

[0020] in, represent the echo signal and additive white Gaussian noise received by the array, represents the array flow matrix, represents the complex amplitude of the echo signal, K represents the number of far-field sources, and the steering vector represented by each column of Γ is expressed as:

[0021]

[0022] n=0,···,N-1 (3)

[0023] Among them, d n represents the distance between the nth array element and the reference array element. The reference array element is usually the first or last array element of the linear array. λ is the wavelength of the carrier signal, and θ k represents the kth target arrival direction to be estimated, [·] T stands for transpose. For a uniform linear array, the distance between the nth element and the reference element is expressed as follows:

[0024] d n =nd (4)

[0025] Where d represents the array element spacing, which is usually λ / 2.

[0026] Specifically, for a half-wavelength uniform linear array with N elements, if its echo There are K targets in , and N>K, then the narrowband single snapshot echo signal y of the nth array element is n Expressed as:

[0027]

[0028] Where k = 1, ···, K, n = 0, ···, N-1, d = λ / 2 represents the array element spacing, λ is the wavelength of the carrier signal, θ k represents the kth target arrival direction to be estimated, is the noise term.

[0029] The inverse beamforming observation model characteristics in step (2) are as follows:

[0030] Let the N-dimensional echo signal y be a discrete sampling function y(n) of length N, and let the ratio of wavelength λ to array element spacing d be G. Then y(n) can be expressed as the sum of K complex exponential single-frequency signals with n as the variable:

[0031]

[0032] f k = sin(θ k ) / G (7)

[0033] Construct N-point discrete inverse Fourier transform matrix Beamforming is implemented by using discrete inverse Fourier transform on y(n). The frequencies of the K signals are estimated based on the positions of the local maxima in the energy spectrum. Finally, the direction of arrival of each target is calculated inversely according to formula (7). The above beamforming process can be expressed in the form of matrix multiplication as follows:

[0034]

[0035] in, represents the sparse vector of energy spectrum for beamforming, is the N-point discrete Fourier transform matrix, since The local maximum position of corresponds to the arrival direction of the target and there are noise and sidelobe effects in the energy spectrum. is the target arrival direction sparse vector x with noise and sidelobe effects The sum of , then:

[0036]

[0037] Performing discrete Fourier transform on both ends of formula (9) yields the inverse beamforming observation model based on discrete Fourier transform:

[0038]

[0039] Assuming the normalized discrete sampling frequency of the N array element echo signals is 1 Hz, the spectral resolution and accuracy of beamforming achieved by discrete inverse Fourier transform are both 1 / N Hz.

[0040] Step (3) is as follows:

[0041] Assume that there is a half-wavelength uniform linear array with P (P>N) array elements, and its single snapshot echo is given by Indicates that using y P and the discrete Fourier transform F at point P P Construct an inverse beamforming observation model:

[0042]

[0043] Among them, the direction of arrival sparse vector The spectral resolution and accuracy is .y 1 / P Hz.

[0044] Step (4) is specifically as follows:

[0045] To increase the spectral resolution and accuracy of the direction of arrival (DOA), the number of array signal sampling points and the number of discrete inverse Fourier transform (DIFT) points must be increased simultaneously. However, the number of array signal sampling points is equivalent to the number of array elements and cannot be directly increased. Therefore, the present invention proposes to use virtual element downsampling.

[0046] Think y P is the virtual array full sampling data, and the N array element echo signal y is from y P The zero-filling vector constructed by downsampling the first N array data and the downsampling matrix of equal-dimensional arrays As shown below:

[0047]

[0048] Step (5) is specifically as follows:

[0049] Substituting the P-point inverse beamforming observation model (11) into the formula (12), we can obtain:

[0050]

[0051] in, represents the sum of the noise in the downsampled array domain and the inverse beamforming sidelobes. At this time, the sparse vector x of the direction of arrival to be solved in the observation model (13)() is P The spectrum resolution and accuracy are improved to 1 / P Hz.

[0052] Step (6) is specifically as follows:

[0053] The vector x in the inverse beamforming super-resolution observation model is P By adding sparse constraints, we can obtain a super-resolution optimization model based on the weighted L1 norm:

[0054]

[0055] Among them, x P is the direction of arrival super-resolution result to be solved, ||·||2 is the L2 norm, is the weighted L1 norm, x P (i) is x P The i-th element of , wi represents the weight of the i-th element;

[0056] At this time, formula (14) is a sparse optimization problem, which can be quickly solved using MATLAB's CVX toolbox. The CVX solution result is:

[0057]

[0058] in, represents the final direction-of-arrival super-resolution result, and CVX(·) represents the CVX toolbox solution process.

[0059] Beneficial effects of the present invention:

[0060] In view of the influence of array number and array position on DOA estimation, the present invention re-analyzes beamforming (DBF) from the perspective of frequency analysis. The present invention proposes a super-resolution DOA estimation method for uniform linear array based on inverse beamforming. This method is based on beamforming and constructs a super-resolution DOA observation model based on discrete Fourier transform, ultimately achieving super-resolution DOA estimation of linear array. By comparing the DOA estimation method based on discrete Fourier transform matrix F in step 2, the DOA estimation method based on discrete Fourier transform matrix F is obtained. N The inverse beamforming observation model shows that this observation model maintains the matrix dimension and array domain redundant information, making the related optimization problem easy to solve. Due to the use of a DBF and FFT-based model construction method, the proposed super-resolution observation model is universally applicable and easy to implement. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 It is a schematic diagram of the process of the present invention.

[0062] Figure 2 This is a diagram showing the principle verification and effect comparison of the uniform linear array super-resolution direction of arrival estimation method with inverse beamforming provided by the present invention.

[0063] Figure 3 for Figure 2 Schematic diagram of the enlarged results of the effect comparison chart. DETAILED DESCRIPTION

[0064] The present invention will be described in further detail below with reference to the accompanying drawings.

[0065] Please refer to Figure 1 and Figure 2 ,in, Figure 1 A schematic diagram of a uniform linear array super-resolution direction-of-arrival estimation method based on inverse beamforming provided by the present invention;

[0066] The method for super-resolution direction of arrival estimation of a uniform linear array based on inverse beamforming includes the following steps:

[0067] Step 1: Construct a narrowband single snapshot echo signal model of the linear array;

[0068] Assuming a linear array with N elements, its narrowband single snapshot echo signal model can be expressed as

[0069] y=Γs+e noise (1)

[0070] in, represent the echo signal and additive white Gaussian noise received by the array, represents the array flow matrix, represents the complex amplitude of the echo signal, and K represents the number of far-field sources. For the steering vector represented by each column of Γ, it can be expressed as:

[0071]

[0072] n=0,···,N-1 (3)

[0073] Among them, d n Indicates the distance between the nth array element and the reference array element, which is usually the first or last array element of the linear array. λ is the wavelength of the carrier signal, θ k Indicates the kth target arrival direction to be estimated. [·] T Represents transpose. For a uniform linear array, the distance between the nth element and the reference element can be expressed as follows:

[0074] d n =nd (4)

[0075] Where d is the element spacing, which is usually λ / 2 to improve the accuracy of the DOA estimation. For a non-uniform linear array, the distance between the nth element and the reference element is Given by a random increasing sequence.

[0076] Specifically, for a half-wavelength uniform linear array with N elements, if its echo There are K targets in , and generally N>K, then the narrowband single snapshot echo signal y of the nth array element is n It can be expressed as:

[0077]

[0078] Where k = 1, ···, K, n = 0, ···, N-1, d = λ / 2 represents the array element spacing, λ is the wavelength of the carrier signal, θ k represents the kth target arrival direction to be estimated, is the noise term.

[0079] Step 2: Based on beamforming, construct a discrete Fourier transform matrix F N The inverse beamforming observation model.

[0080] Let the N-dimensional echo vector y be a discrete sampling function y(n) of length N, and let the ratio of wavelength λ to array element spacing d be G. Then y(n) can be expressed as the sum of K complex exponential single-frequency signals with n as the variable:

[0081]

[0082] f k = sin(θ k ) / G (7)

[0083] Construct N-point discrete inverse Fourier transform matrix Beamforming is achieved by using the discrete inverse Fourier transform on y(n). The frequencies of the K signals are estimated based on the local maximum position in the energy spectrum. Finally, the direction of arrival of each target is calculated based on formula (7). The above beamforming process can be expressed in matrix multiplication form as follows:

[0084]

[0085] in, represents the sparse vector of energy spectrum for beamforming, is the N-point discrete Fourier transform matrix. The local maximum position of corresponds to the arrival direction of the target and there are noise and sidelobe effects in the energy spectrum. is the target arrival direction sparse vector x with noise and sidelobe effects The sum of , then:

[0086]

[0087] Performing discrete Fourier transform on both ends of formula (9) yields the inverse beamforming observation model based on discrete Fourier transform:

[0088]

[0089] Assuming the normalized discrete sampling frequency of the N array element echo signals is 1 Hz, the spectral resolution and accuracy of the beamforming achieved by discrete inverse Fourier transform are both 1 / N Hz.

[0090] Step 3: Assume that there are P (P>N) array elements with full sampling data y P , construct the discrete Fourier transform matrix F P The inverse beamforming observation model.

[0091] Assume that there is a half-wavelength uniform linear array with P (P>N) array elements, and its single snapshot echo is given by Indicates. Use y P and the discrete Fourier transform F at point P P Construct an inverse beamforming observation model:

[0092]

[0093] Among them, the direction of arrival sparse vector The spectral resolution and accuracy is 1 / P Hz.

[0094] Step 4: Using the idea of ​​virtual array element downsampling, construct the downsampling matrix Θ, N actual array data y NP It can be regarded as using Θ to sample the full data y of P virtual array elements P The result of downsampling.

[0095] To increase the spectral resolution and accuracy of the direction of arrival (DOA), the number of array signal sampling points and the number of discrete inverse Fourier transform (DIFT) points must be increased simultaneously. However, the number of array signal sampling points is equivalent to the number of array elements. Given a fixed number of elements, the number of array signal sampling points cannot be directly increased. Therefore, the present invention proposes the concept of virtual element downsampling.

[0096] It can be considered that y P is the virtual array full sampling data, and the N array element echo signal y is from y P The first N array data are downsampled. Construct the zero-filled vector of y and the downsampling matrix of equal-dimensional arrays As shown below:

[0097]

[0098] Step 5: Based on the inverse beamforming observation model at point P, according to Θ, F P and y NP , construct a P (P>N) point inverse beamforming super-resolution observation model.

[0099] Substituting the inverse beamforming observation model (11) at point P described in step 3 into the formula (12) described in step 4, we can obtain:

[0100]

[0101] in, represents the sum of the noise in the downsampled array domain and the inverse beamforming sidelobes. At this time, the sparse vector x of the direction of arrival to be solved in the observation model (13) is P The spectrum resolution and accuracy are improved to 1 / P Hz.

[0102] Step 6: Build a super-resolution optimization model based on the weighted L1 norm, use MATLAB's CVX toolbox to solve the optimization model, and finally achieve super-resolution estimation of the direction of arrival.

[0103] The vector x in the inverse beamforming super-resolution observation model described in step 5 is P By adding sparse constraints, we can obtain a super-resolution optimization model based on the weighted L1 norm:

[0104]

[0105] Among them, x P is the direction of arrival super-resolution result to be solved, ||·||2 is the L2 norm, is the weighted L1 norm, x P (i) is x P The i-th element of , wi represents the weight of the i-th element.

[0106] At this time, formula (14) is a sparse optimization problem, which can be quickly solved using MATLAB's CVX toolbox. The CVX solution result is:

[0107]

[0108] in, represents the final direction-of-arrival super-resolution result, and CVX(·) represents the CVX toolbox solution process.

[0109] like Figure 2 As shown in the figure, a uniform 16-element linear array with half-wavelength spacing was used in the experiment. The dual-target echo signal-to-noise ratio was 15dB. A 64-point discrete Fourier transform was used to construct an inverse beamforming super-resolution direction-of-arrival observation model. Based on this, an optimization model with a weighted L1 norm as a sparse constraint was designed. Finally, the optimization model was solved using MATLAB's CVX toolbox to achieve super-resolution direction-of-arrival estimation. The theoretical resolution of the direction-of-arrival estimation of 16 elements is about 6.77°, while the angles of the dual targets in the experiment were -1.79° and 1.79°, respectively. Figure 2 and Figure 3Experimental results show that the method proposed in this application can achieve super-resolution direction of arrival estimation. This paper proposes a super-resolution direction of arrival estimation method for uniform linear arrays based on inverse beamforming. This method, based on beamforming, constructs a super-resolution direction of arrival observation model based on discrete Fourier transform, ultimately achieving super-resolution direction of arrival estimation for linear arrays.

Claims

1. A uniform linear array super-resolution direction of arrival estimation method based on inverse beamforming, characterized in that: The following steps are included: Step (1): Construct a narrowband single snapshot echo signal model of the linear array, implement the signal model for direction of arrival estimation, and form a beam; Step (2): Based on beamforming, construct a discrete Fourier transform matrix F N The inverse beamforming observation model of Step (3): Assume that there are P array elements with full sampling data y P , P>N, based on the observation model, full sampling data expansion is performed to construct a discrete Fourier transform matrix F P The inverse beamforming observation model of Step (4): Use virtual array element downsampling to construct the downsampling matrix Θ, N actual array element data y NP Considered as the full sampling data y of P virtual array elements using Θ P The result of downsampling; Step (5): Based on the inverse beamforming observation model at point P in step (3), use Θ, F P and y NP , construct the P-point inverse beamforming super-resolution observation model; Step (6): Construct a super-resolution optimization model based on the weighted L1 norm, use MATLAB's CVX toolbox to solve the optimization model, and finally achieve super-resolution estimation of the direction of arrival.

2. The method for uniform linear array super-resolution direction of arrival estimation based on inverse beamforming according to claim 1, characterized in that: Step (1) is specifically as follows: Assume a linear array with N elements, and its narrowband single snapshot echo signal model is expressed as y=Γs+e noise (1) in, represent the echo signal and additive white Gaussian noise received by the array, represents the array flow matrix, represents the complex amplitude of the echo signal, K represents the number of far-field sources, and the steering vector represented by each column of Γ is expressed as: n=0,···,N-1 (3) Among them, d n represents the distance between the nth array element and the reference array element. The reference array element is the first or last array element of the linear array. λ is the wavelength of the carrier signal, and θ k represents the kth target arrival direction to be estimated, [·] T stands for transpose. For a uniform linear array, the distance between the nth element and the reference element is expressed as follows: d n =nd (4) Where d represents the array element spacing, which is λ / 2.

3. The method for uniform linear array super-resolution direction of arrival estimation based on inverse beamforming according to claim 2, characterized in that: For a half-wavelength uniform linear array with N elements, if its echo There are K targets in , and N>K, then the narrowband single snapshot echo signal y of the nth array element is n Expressed as: Where k = 1, ···, K, n = 0, ···, N-1, d = λ / 2 represents the array element spacing, λ is the wavelength of the carrier signal, θ k represents the kth target arrival direction to be estimated, is the noise term.

4. The method for uniform linear array super-resolution direction of arrival estimation based on inverse beamforming according to claim 3, characterized in that: The inverse beamforming observation model characteristics in step (2) are as follows: Let the N-dimensional echo signal y be a discrete sampling function y(n) of length N, and let the ratio of wavelength λ to array element spacing d be G. Then y(n) is expressed as the sum of K complex exponential single-frequency signals with n as the variable: f k =sin(θ k ) / G (7) Construct N-point discrete inverse Fourier transform matrix Beamforming is implemented by using discrete inverse Fourier transform on y(n). The frequencies of the K signals are estimated based on the positions of the local maxima in the energy spectrum. Finally, the direction of arrival of each target is calculated inversely according to formula (7). The above beamforming process can be expressed in the form of matrix multiplication as follows: in, represents the sparse vector of energy spectrum for beamforming, is the N-point discrete Fourier transform matrix, is the target arrival direction sparse vector x with noise and sidelobe effects The sum of , then: Performing discrete Fourier transform on both ends of formula (9) yields the inverse beamforming observation model based on discrete Fourier transform: Assuming the normalized discrete sampling frequency of the N array element echo signals is 1 Hz, the spectral resolution and accuracy of beamforming achieved by discrete inverse Fourier transform are both 1 / N Hz.

5. The method for uniform linear array super-resolution direction of arrival estimation based on inverse beamforming according to claim 1, characterized in that: Step (3) is as follows: Assume that there is a half-wavelength uniform linear array with P elements, and its single snapshot echo is given by Indicates that using y P and the discrete Fourier transform F at point P P Construct an inverse beamforming observation model: Among them, the direction of arrival sparse vector The spectral resolution and accuracy is 1 / P Hz.

6. The method for uniform linear array super-resolution direction of arrival estimation based on inverse beamforming according to claim 5, characterized in that: Step (4) is specifically as follows: Think y P is the virtual array full sampling data, and the N array element echo signal y is from y P The zero-filling vector constructed by downsampling the first N array data and the downsampling matrix of equal-dimensional arrays As shown below:

7. The method for uniform linear array super-resolution direction of arrival estimation based on inverse beamforming according to claim 6, characterized in that: Step (5) is specifically as follows: Substituting the P-point inverse beamforming observation model (11) into the formula (12), we can obtain: in, represents the sum of the noise in the downsampled array domain and the inverse beamforming sidelobes. At this time, the sparse vector x of the direction of arrival to be solved in the observation model (13) is P The spectrum resolution and accuracy are improved to 1 / P Hz.

8. The method for uniform linear array super-resolution direction of arrival estimation based on inverse beamforming according to claim 1, characterized in that: Step (6) is specifically as follows: The vector x in the inverse beamforming super-resolution observation model is P By adding sparse constraints, we can obtain a super-resolution optimization model based on the weighted L1 norm: Among them, x P is the direction of arrival super-resolution result to be solved, ||·||2 is the L2 norm, is the weighted L1 norm, x P (i) is x P The i-th element, w i represents the weight of the i-th element; At this time, formula (14) is a sparse optimization problem, which can be quickly solved using MATLAB's CVX toolbox. The CVX solution result is: in, represents the final direction-of-arrival super-resolution result, and CVX(·) represents the CVX toolbox solution process.

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