Robust Superdirective Beam Optimization Method of Arbitrary Order Applicable to Ring Array

Abstract

The present invention relates to an arbitrary-order robust superdirective beam optimization method applicable to circular-ring arrays. Based on the block-circulant property of the noise correlation matrix of circular-ring arrays, eigenvalue decomposition is performed, and the optimal weight vector in the element domain can be transformed into the optimal weight vector in the eigenbeam domain. Then, a coefficient is added in front of the highest-order integer-order weight vector, and by adjusting this coefficient, arbitrary-order superdirectivity can be obtained. The method disclosed by the present invention overcomes the deficiencies of only being able to obtain integer-order superdirectivity or only being able to obtain the fractional-order superdirectivity index by using the desired beam fitting. A more flexible arbitrary-order superdirective beam is obtained. Compared with the methods disclosed in the prior art, it is an arbitrary-order robust superdirective beam optimization method applicable to circular-ring arrays with arbitrary-form subarrays, with less computational complexity and a wider application range.

Description

Technical Field

[0001] The present invention belongs to the field of beamforming methods, and relates to a method for optimizing arbitrary-order robust superdirective beams applicable to circular-ring arrays, which is applicable to low signal-to-noise ratio target detection and high-resolution estimation of target azimuth for arbitrary arrays, and belongs to the fields of acoustics, array signal processing, sonar technology, etc. Background Art

[0002] Beamforming is a technology that weights the data received by a sensor array to make it directional, so that it can suppress noise and interference while keeping the desired signal unchanged. Among them, the superdirective beamforming method has received extensive attention and in-depth research due to its better performance in directivity, directivity index, and array miniaturization. Document 1 "Theoretical and practical solutions for high-order superdirectivity of circular sensor arrays. IEEE Trans. Ind. Electron., 2013, 60(1): 203-209." discloses a superdirective method applicable to circular arrays based on eigenbeam decomposition and synthesis. Through rank reduction processing, robust superdirectivity can be obtained, but only integer-order superdirectivity can be obtained; Document 2 "A method for forming arbitrary-order superdirective beams of a circular array, CN202010142632.7" discloses an arbitrary-order superdirective method based on a circular array. This method introduces fractional orders between the original integer orders, more flexibly realizes robust supergain, obtains the optimal result consistent with numerical calculation, and derives a closed-form weight expression, but this method is only applicable to circular arrays; Document 3 "Design of Planar Differential Microphone Arrays With Fractional Orders. IEEE / ACM Trans. Audio Speech Lang. Process., 2020, 28: 116-130" discloses a method for designing fractional-order differential microphone arrays, which can obtain continuous directivity indices. This method needs to obtain fractional-order directivity by fitting the desired beam, and the desired beam is obtained by combining the differential beams of two adjacent integer orders, but this method needs to use the desired beam and is only applicable to planar arrays, and it is difficult to play a role in volume arrays. Summary of the Invention

[0003] Technical Problems to be Solved

[0004] To avoid the deficiencies of the prior art, the present invention proposes a method for optimizing arbitrary-order robust superdirective beams applicable to circular-array-like arrays, avoiding the deficiencies of the prior art that can only obtain integer-order superdirectivity or can only obtain fractional-order superdirectivity indices by fitting the desired beam.

[0005] Technical solution

[0006] A method for optimizing arbitrary-order robust superdirective beams applicable to circular-array-like arrays, characterized by the following steps:

[0007] Step 1: Calculate the weight vector ω of the arbitrary-order N' superdirective beam, where the arbitrary-order N' is N' = N + η, 0 ≤ η ≤ 1, N' is a real number and 0 ≤ N' ≤ M - 1; N is the largest non-negative integer less than N', and when N' = 0, N = 0, η = 0, and the weight vector ω is:

[0008]

[0009] where: Ω0 = (θ0, φ0) is the main lobe direction of the beam, λ represents the wavelength of the incident plane wave, M is the total number of array elements;

[0010] E(Ω) = U H p(Ω), R n = UΓU H , (·) H denotes conjugate transpose, (·) T denotes transpose; p(Ω) = exp(-iW T k) is the unit-amplitude plane wave signal received by the array from the direction Ω = (θ, φ);

[0011] In the formula: k = -k[sinθcosφ, sinθsinφ, cosθ] T , φ is the azimuth angle, θ is the elevation angle, k = 2π / λ;

[0012] W = [Y0, Y1, …, Y G-1 is a 3×M-dimensional matrix composed of all array element position vectors, where g = 0, 1, …, G - 1, Y g = [r g,0 , r g,1 , …, r g,L-1 represents a 3×L-dimensional matrix composed of the position vectors of all array elements in the g-th subarray, where l = 0, 1, …, L - 1; G is the number of subarrays on the circle, L is the number of array elements in each subarray, and the total number of array elements M = G×L;

[0013] r g,l represents the position vector of the l-th element in the g-th sub-array, r g,l is a 3×1 dimensional column vector

[0014]

[0015] where: d m1m2 is the distance between the m1-th element and the m2-th element, and R n is partitioned;

[0016] The said R n is:

[0017]

[0018] where each b g (g = 0, 1, …, G - 1) is an L×L matrix, where A g is composed of all the eigenvectors of H g , and Γ g is the eigenvalue diagonal matrix of H g ;

[0019] U is composed of all the eigenvectors of R n , and the expression is U = [V0; V1; …; …; V G-1 , where Γ is the eigenvalue diagonal matrix of R n , and the expression is Γ = diag(Γ g ), diag(·) represents the diagonal matrix, and all the eigenvectors of R n correspond to eigenvalues satisfying λ M-1 ≥λ M-2 ≥…≥λ0;

[0020] Step 2: Synthesize and optimize the N'-th order superdirective beam, and calculate the corresponding directivity index and error sensitivity index:

[0021] The synthesis of the N'-th order superdirective beam B(Ω) is:

[0022]

[0023] The said

[0024] The directivity index D of the N'-th order superdirective is calculated by the following formula:

[0025]

[0026] The said Λ n= diag{λ0, λ1, …, λ N , λ N+1}, |·| represents taking the absolute value, and the directivity index is DI = 10log 10 D;

[0027] The error sensitivity function T of any-order N' superdirectivity is calculated by the following formula:

[0028] T = ||ω|| 2

[0029] The ||·|| represents calculating the 2-norm of the vector, and the error sensitivity index is SI = 10log 10 T.

[0030] Advantageous Effects

[0031] A method for optimizing any-order robust superdirective beams applicable to circular-ring arrays proposed by the present invention performs eigenvalue decomposition based on the block-circulant property of the noise correlation matrix of circular-ring arrays, and can transform the optimal weight vector in the element domain into the optimal weight vector in the eigenbeam domain. Then, by adding a coefficient before the highest-order integer-order weight vector, adjusting this coefficient can obtain superdirectivity of any order. The method disclosed by the present invention overcomes the deficiencies of only being able to obtain integer-order superdirectivity or only being able to obtain the fractional-order directivity index by using the desired beam fitting. By using the property that the optimal weight vector in the element domain can be transformed into the optimal weight vector in the eigenbeam domain, a control parameter is added before the highest integer-order weight coefficient selected in the eigenbeam domain, and combined with all the weight coefficients before the adjacent integer orders, a weight vector that can achieve superdirectivity of any order is formed. A more flexible superdirective beam of any order is obtained. Compared with the methods disclosed in the prior art, it is an any-order robust superdirective beam optimization method applicable to circular-ring arrays of any form of subarray, with less computational complexity and a wider application range.

[0032] The weight vector of the method of the present invention is composed of the highest integer-order weight coefficient with a control parameter added in the eigenbeam domain and all the weight coefficients before its adjacent integer orders. By adjusting this parameter, the synthesis of superdirective beams of any order including integer orders can be achieved, and superdirective beams that cannot be obtained by using the method disclosed in Document 1 can be obtained.

[0033] The method of the present invention is applicable to circular-ring arrays of any subarray, and has a wider application range compared with the method disclosed in Document 2.

[0034] The weight vector of the method of the present invention is directly obtained from the optimal weight vector. Compared with the method disclosed in Document 3, it does not need to use any desired beam and is applicable to volume arrays. Description of the Drawings

[0035] Figure 1It is a schematic diagram of a double-layer cylindrical array using a 2-element linear array as a sub-array for Simulation 1. Figure 1 (a) is a three-dimensional view. Figure 1 (b) is a top-down plan view.

[0036] Figure 2 It is the directivity index and error sensitivity index of Simulation 1 at different given orders. Figure 2 (a) is the directivity index. Figure 2 (b) is the error sensitivity index.

[0037] Figure 3 It is the directivity index, error sensitivity index, and broadband beam pattern of Simulation 1 when different orders are selected at a given frequency of 500 Hz. Figure 3 (a) is the directivity index. Figure 3 (b) is the error sensitivity index. Figure 3 (c) is the broadband beam pattern.

[0038] Figure 4 It is a schematic diagram of a frustum array using a 3-element linear array as a sub-array for Simulation 2. Figure 4 (a) is a three-dimensional view. Figure 4 (b) is a top-down plan view.

[0039] Figure 5 It is the directivity index and error sensitivity index of Simulation 2 at different given orders. Figure 5 (a) is the directivity index. Figure 5 (b) is the error sensitivity index.

[0040] Figure 6 It is the directivity index, error sensitivity index, and broadband beam pattern of Simulation 2 when different orders are selected at a given frequency of 300 Hz. Figure 6 (a) is the directivity index. Figure 6 (b) is the error sensitivity index. Figure 6 (c) is the broadband beam pattern.

[0041] Figure 7 It is a schematic diagram of a cylindrical array using a cylindrical array as a sub-array for Simulation 3. Figure 7 (a) is a three-dimensional view. Figure 7 (b) is a top-down plan view.

[0042] Figure 8 It is the directivity index and error sensitivity index of Simulation 3 at different given orders. Figure 8 (a) is the directivity index. Figure 8 (b) is the error sensitivity index.

[0043] Figure 9They are the directivity index, error sensitivity index, and broadband beam pattern of Simulation 3 when different orders are selected at a given frequency of 250 Hz. Figure 9 (a) is the directivity index, Figure 9 (b) is the error sensitivity index, Figure 9 (c) is the broadband beam pattern.

[0044] Figure 10 It is the sensor received signal model Specific implementation manners

[0045] The present invention will be further described below in conjunction with embodiments and the accompanying drawings:

[0046] The following implementation manners will help those skilled in the art to further understand the present invention, but do not limit the present invention in any form. It should be noted that for those skilled in the art, without departing from the concept of the present invention, several changes and improvements can still be made. These all belong to the protection scope of the present invention:

[0047] Step 1: Calculate the weight vector ω of the arbitrary-order N' superdirective beam:

[0048]

[0049] The arbitrary order N' is N' = N + η, 0 ≤ η ≤ 1, N' is a real number and 0 ≤ N' ≤ M - 1; N is the largest non-negative integer less than N', and when N' = 0, N = 0 and η = 0;

[0050] The E(Ω) = U H p(Ω), R n = UΓU H , M is the total number of array elements, Ω0 = (θ0, φ0) is the main lobe direction of the beam, (·) H represents the conjugate transpose, (·) T represents the transpose;

[0051] p(Ω) is the unit amplitude plane wave signal incident on the array from the direction Ω = (θ, φ), and the expression is:

[0052] p(Ω) = exp(-iW T k)

[0053] In the formula: k = -k[sinθcosφ, sinθsinφ, cosθ] T , φ is the azimuth angle, θ is the elevation angle, k = 2π / λ, λ represents the wavelength of the incident plane wave, and W is a 3×M-dimensional matrix composed of all array element position vectors, and its expression is W = [Y0, Y1,..., Y G-1, where \(g = 0, 1, \ldots, G - 1\), \(Y\) g represents a \(3\times L\) - dimensional matrix composed of the position vectors of all array elements in the \(g\) - th sub - array, \(Y\) g =\([r\) g,0 , \(r\) g,1 , \(\ldots\), \(r\) g,L-1 , where \(l = 0, 1, \ldots, L - 1\), \(r\) g,l represents the position vector of the \(l\) - th array element in the \(g\) - th sub - array, \(r\) g,l is a \(3\times1\) - dimensional column vector, \(G\) is the number of sub - arrays on the ring, \(L\) is the number of array elements in each sub - array, and the total number of array elements \(M = G\times L\).

[0054] In the formula: \(R\) n is the normalized noise covariance matrix, which has the property of block - circulant, and the expression is:

[0055]

[0056] where: is the distance between the \(m_1\) - th array element and the \(m_2\) - th array element. Block - divide \(R\) n , then \(R\) n can be expressed as:

[0057]

[0058] where each \(b\) g (\(g = 0, 1, \ldots, G - 1\)) is an \(L\times L\) matrix. where \(A\) g is composed of all the eigen - vectors of \(H\) g , \(\Gamma\) g is the diagonal matrix of the eigenvalues of \(H\) g , and it needs to be obtained through numerical calculation.

[0059] In the formula, \(U\) is composed of all the eigen - vectors of \(R\) n , and the expression is \(U=[V_0; V_1; \ldots; \ldots; V\) G-1 , where \(\Gamma\) is the diagonal matrix of the eigenvalues of \(R\) n , and the expression is \(\Gamma=\text{diag}(\Gamma\) g ), \(\text{diag}(\cdot)\) represents the diagonal matrix. Note that all the eigen - vectors of \(R\) n should satisfy the corresponding eigenvalues \(\lambda\) M-1 \(\geq\lambda\) M-2 \(\geq\ldots\geq\lambda_0\).

[0060] Step 2: Synthesize an arbitrary - order \(N'\) super - directive beam, and calculate the corresponding directivity index and error sensitivity index:

[0061] The superdirective beam B(Ω) of any order N' is synthesized by the following formula:

[0062]

[0063] It is described that:

[0064]

[0065] The directivity index D of the superdirectivity of any order N' is calculated by the following formula:

[0066]

[0067] It is described that

[0068] Λ n = diag{λ0, λ1, …, λ N , λ N+1}}, |·| represents taking the absolute value, and the directivity index is DI = 10log 10 D;

[0069] The error sensitivity function T of the superdirectivity of any order N' is calculated by the following formula:

[0070] T = ||ω|| 2

[0071] It is described that ||·|| represents calculating the 2-norm of a vector, and the error sensitivity index is SI = 10log 10 T.

[0072] Specific simulation example:

[0073] Simulation 1. Consider a double-layer cylindrical array. The diameter of each circular ring layer is 1 meter, the number of array elements in each layer is 8, the total number of array elements M = 16, and the array element position coordinates are as Figure 1 shown. The incident direction of the plane wave is The frequency range is [10 Hz, 1200 Hz], and the sound speed is 1500 m / s.

[0074] For the double-layer cylindrical array considered in Simulation 1, G = 8, L = 2;

[0075] In particular, when the array is a double-layer cylindrical array, not only the normalized noise covariance matrix R n is a block circulant matrix, but each block of R n is also a circulant matrix. Therefore, H g is also a block circulant matrix. Define the angle between the center of each circular ring and two array elements as β = 2π / G. Then the eigenvalue expression of H g is λ g,l = (H g(1,1) + H g(1,2) )e islβ, the eigenvector expression is: v g,l = L -1 / 2 [1, e ilβ T , where λ g,l and v g,l respectively represent the l-th eigenvalue of H g and its corresponding eigenvector, H g(1,1) and H g(1,2) represent the two elements in the first row of H g , g = 0, 1, …, G - 1;

[0076] As Figure 2 shown, arbitrary orders N' = 5.9, 7.5, and 11.1 are set. As the frequency increases, the superdirectivity index and error sensitivity index of different orders both decrease, meaning the directivity decreases but the robustness increases. The higher the order, the higher the directivity of the directivity method, and the wider the frequency band where the superdirectivity index of the superdirectivity method is greater than the directivity index of the DAS method, meaning superdirectivity can be obtained in a larger frequency range.

[0077] As Figure 3 shown, arbitrary order N' is set to vary continuously from 0 to 15, the frequency is 500 Hz, and the wavelength λ = 3 m. At this time, the curves of the directivity index and error sensitivity index varying with N' are as Figure 3 (a), (b) shown. As N' varies continuously from 0 to 15, the directivity index and error sensitivity index can also take continuous values, different from the integer-order superdirectivity that can only obtain discrete values. At the same time, both increase as the order N' increases, meaning the directivity increases with the increase of the order, while the robustness becomes worse. The directivity index and error sensitivity index both show a certain degree of non-smoothness at some integer orders, but they are still continuous. In addition, the directivity index of the conventional method is approximately the superdirectivity result around N' = 0.5, so it can be considered that the order of the conventional method is close to 0.5.

[0078] Simulation 2: Consider a three-layer frustum array with a total number of array elements M = 18, and the array element position coordinates are as Figure 4 shown, and the diameters of each layer of the ring are 1 m, 1.2 m, and 1.4 m respectively. The incident direction of the plane wave is the frequency range is [10 Hz, 600 Hz], and the sound speed is 1500 m / s.

[0079] For the three-layer frustum array considered in Simulation 2, G = 6, L = 3;

[0080] As Figure 5 shown, arbitrary orders N' = 7.5, 8.7, and 15.9 are set. Figure 5 ​All orders of superdirective beams are superior to those of the DAS method and smaller than those of the MVDR method. As the frequency increases, the superdirectivity index and error sensitivity index for different orders both decrease, meaning that the directivity decreases but the robustness increases. The higher the order, the wider the frequency band where the superdirectivity method index is greater than the DAS method directivity index, meaning that superdirectivity can be obtained over a larger frequency range.

[0081] As Figure 6 shown, setting any order N' to vary continuously from 0 to 17, with a frequency of 300 Hz and a wavelength λ = 5 m, the curves of the directivity index and error sensitivity index varying with N' are as Figure 6 (a) and (b) shown. As N' varies continuously from 0 to 17, the directivity index and error sensitivity index can also take continuous values, different from the integer-order superdirectivity that can only obtain discrete values. At the same time, both increase as the order N' increases, meaning that the directivity increases with the increase of the order while the robustness deteriorates. The directivity index and error sensitivity index both show a certain lack of smoothness at integer orders but are still continuous. In addition, the directivity index of the conventional method is approximately the same as the superdirectivity result at about N' = 0.3, so it can be considered that the order of the conventional method is close to 0.3.

[0082] Simulation 3: Consider a cylindrical array with double-layer cylindrical arrays as sub-arrays, consisting of 4 cylindrical sub-arrays, and each sub-array consists of 2 4-element circular arrays. Then the total number of array elements M = 32, and the array element position coordinates are as Figure 7 shown. The incident direction of the plane wave is The frequency range is [10 Hz, 800 Hz], and the sound speed is 1500 m / s.

[0083] For the array considered in Simulation 3, G = 4, L = 8;

[0084] As Figure 8 shown, setting any order N' = 10.8, 23.9, and 28.7. As the frequency increases, the superdirectivity index and error sensitivity index for different orders generally both decrease, meaning that the directivity decreases but the robustness increases. The higher the order, the wider the frequency band where the superdirectivity method index is greater than the DAS method directivity index, meaning that superdirectivity can be obtained over a larger frequency range. When the frequency is greater than 600 Hz, the directivity index of the conventional method exceeds that of the superdirectivity method with an order of 10.8, indicating that the beam of this order no longer has superdirectivity when the frequency is greater than 600 Hz.

[0085] As Figure 9 shown, setting any order N' to vary continuously from 0 to 31, with a frequency of 250 Hz and a wavelength λ = 6 m, the curves of the directivity index and error sensitivity index varying with N' are as Figure 9(a) and (b) as shown. As N' continuously changes from 0 to 31, the directivity index and the error sensitivity index can also take continuous values, which is different from the integer-order superdirectivity that can only obtain discrete values. At the same time, both increase as the order N' increases, meaning that the directivity increases with the increase of the order, while the robustness deteriorates. The directivity index and the error sensitivity index both show a certain degree of non-smoothness at some integer orders, but they are still continuous. In addition, the directivity index of the conventional method is approximately the same as the superdirectivity result around N' = 0.7, so it can be considered that the order of the conventional method is close to 0.7.

[0086] In summary, the method of the present invention can obtain superdirective beams of any order and the corresponding directivity index and error sensitivity index, and has better flexibility compared with the integer-order superdirectivity method. At the same time, the method disclosed in the present invention is applicable to the formation of superdirective beams of any order for any form of subarray circular arrays, with less computational complexity, a wider application range, and no additional desired beam is required.

Claims

1. A method for optimizing arbitrary-order robust superdirective beams applicable to circular-array-like arrays, characterized in that The steps are as follows: Step 1: Calculate the weight vector ω of the superdirective beam of any order N', where the any order N' is N' = N + η, 0 ≤ η ≤ 1, N' is a real number and 0 ≤ N' ≤ M - 1; the N is the largest non-negative integer less than N', and when N' = 0, N = 0, η = 0, and the weight vector ω is: where: Ω0 = (θ0, φ0) is the direction of the main lobe of the beam, and λ represents the wavelength of the incident plane wave. M is the total number of array elements; The E(Ω) = U H p(Ω), R n = UΓU H , (·) H denotes conjugate transpose, (·) T denotes transpose; p(Ω) = exp(-iW T k) is the unit amplitude plane wave signal incident from direction Ω = (θ, φ) received by the array; Where: k = -k[sinθcosφ, sinθsinφ, cosθ] T , φ is the azimuth angle, θ is the elevation angle, and k = 2π / λ; W = [Y0, Y1, …, Y G-1 is a 3×M-dimensional matrix composed of all element position vectors, where g = 0, 1, …, G - 1, Y g = [r g,0 , r g,1 , …, r g,L-1 represents a 3×L-dimensional matrix composed of the position vectors of all elements in the g-th subarray, where l = 0, 1, …, L - 1; G is the number of subarrays on the ring, L is the number of elements in each subarray, and the total number of elements M = G×L; r g,l represents the position vector of the \(l\)-th element in the \(g\)-th subarray, \(r\) g,l is a 3×1 dimensional column vector Wherein: is the distance between the m1-th array element and the m2-th array element, and R n is partitioned; The said R n is: where each b g (g = 0, 1, …, G - 1) is an L×L matrix, ρ g = e i2πg / G , where A g is composed of all the eigenvectors of H g , and Γ g is the diagonal matrix of the eigenvalues of H g ; U consists of all the eigenvectors of R n and is expressed as U = [V0; V1; …; …; V G-1 , where Γ is the diagonal matrix of the eigenvalues of R n and is expressed as Γ = diag(Γ g ), diag(·) represents a diagonal matrix, and the eigenvalues corresponding to all the eigenvectors of R n satisfy λ M-1 ≥ λ M-2 ≥ … ≥ λ0; Step 2: Synthesize and optimize the superdirective beam of any order N', and calculate the corresponding directivity index and error sensitivity index: The synthesis of the superdirective beam B(Ω) of any order N' is: The said The directivity index D of the superdirectivity of any order N' is calculated by the following formula: The said Λ n = diag{λ0, λ1, …, λ N , λ N+1}, |·| represents taking the absolute value, and the directivity index is DI = 10log 10 D; The error sensitivity function T of the superdirectivity of any order N' is calculated by the following formula: T = ||ω|| 2 The ||·|| represents the 2-norm of a vector, and the error sensitivity index is SI = 10log 10 T.

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