A three-dimensional beamforming method, device and medium for a small-sized planar array based on acoustic vector hydrophones

Abstract

The present invention provides a three-dimensional beamforming method, device, and medium for a small-sized planar array based on a vector hydrophone. First, the required three-dimensional beam pattern is designed according to requirements, and the designed beam pattern is expanded into a series of spherical harmonic functions to obtain the coefficients of each order of spherical harmonic functions; then, according to the relationship between spherical harmonic functions and multipole modes, each order of spherical harmonic functions is further decomposed into a series of multipole modes to determine the required multipole modes and the coefficients of each multipole mode; afterwards, the finite difference technique is used to extract the required multipole modes, determine the array shape and the coefficients of each receiving sensor; finally, the array receiving channel weighting vectors for synthesizing the desired beam pattern are obtained by combining the obtained coefficients to achieve three-dimensional beamforming. The method of the present invention avoids the vertical amplitude-phase error introduced under the influence of the deep-sea waveguide, and solves the problem that it is difficult for existing low-frequency small-sized arrays to be applied to deep-sea three-dimensional space detection.

Description

Technical Field

[0001] The present invention belongs to the technical field of sonar array signal processing, and in particular relates to a three-dimensional beamforming method, device and medium for a small-sized planar array based on an acoustic vector hydrophone. The present invention relates to a three-dimensional beamforming method for a small-sized acoustic vector planar array in the low frequency range of 10-500 Hz. Background Art

[0002] In the low-frequency and very low-frequency working range, the Rayleigh limit of traditional sonar design seriously restricts the detection performance of underwater sonar arrays, and the breakthrough in the research of small-size arrays with array spacing far less than half a wavelength provides a solution to this problem. In recent years, super-directional beamforming methods based on small-size sonar arrays have become a research hotspot, and a large number of design analyses and computer simulations have demonstrated the advantages of such arrays and their beamforming methods. However, most of these methods are two-dimensional beamforming methods, ignoring the influence of the pitch angle on beamforming, and their performance changes with the change of the steering angle. Usually, such methods are only applicable to application scenarios with small grazing angles such as shallow waters.

[0003] As underwater detection develops into the deep sea, the influence of the pitch angle on the beamforming method increases, and the two-dimensional beamforming method of small-sized arrays gradually cannot meet the detection needs, so it is necessary to study three-dimensional beamforming. However, so far, the research on three-dimensional beamforming algorithms in the field of underwater acoustics has mainly focused on volume arrays. Affected by the ocean waveguide, the amplitude and phase of the sound field in the vertical direction change dramatically, while the sound field and phase changes in the horizontal direction are relatively small. Therefore, in order to meet the needs of deep-sea low-frequency detection, it is necessary to study the three-dimensional beamforming method of small-sized planar arrays. Summary of the invention

[0004] The present invention aims at the contradiction between array performance and array shape and size under the current deep-sea low-frequency detection conditions, and proposes a small-size planar array three-dimensional beamforming method, device and medium based on acoustic vector hydrophone. When the array adjacent spacing d and the signal wavelength λ meet the range of 0.01≤d / λ≤0.2, the method can obtain the required three-dimensional beam with a planar array smaller than the signal wavelength.

[0005] The present invention is implemented by the following technical scheme. The present invention proposes a small-size planar array three-dimensional beamforming method based on acoustic vector hydrophone, which specifically includes the following steps:

[0006] Step 1: Design the required three-dimensional beam pattern, expand the designed three-dimensional beam pattern using spherical harmonic functions in a spherical coordinate system, and determine the coefficients of spherical harmonic functions of each order;

[0007] Step 2: Determine the multipole modes contained in the acoustic vector plane array;

[0008] Step 3: Further expand each order of spherical harmonic functions into the multipole modes possessed by the acoustic vector planar array, and determine the coefficients of each multipole mode;

[0009] Step 4: Use the finite difference technique to extract the required multipole modes, design the required planar array, and determine the coefficients of each receiving sensor when extracting the multipole modes;

[0010] Step 5: Synthesize the obtained coefficients to obtain the weighted vector of the required beamformer, and obtain the designed three-dimensional beam pattern.

[0011] Furthermore, in Step 1,

[0012] Construct a spherical coordinate system with the array center as the origin, design the required three-dimensional beam pattern B(θ, φ), where θ and φ represent the elevation angle and azimuth angle respectively, and expand it with each order of spherical harmonic functions as:

[0013]

[0014] In the above formula, the infinite series expansion is truncated to the Nth order, that is, N represents the highest order of the spherical harmonic functions, where c n,m represents the coefficient after the beam pattern expansion , c = [c 0,0 , c -1,1 ,..., c N,N T is an (N + 1) n,m ×1 vector composed of c 2 coefficients, and the superscript T represents the transpose, is an (N + 1) ×1 column vector composed of spherical harmonic functions of each order with n ≤ N and |m| ≤ n 2 .

[0015] Furthermore, in Step 2,

[0016] From the multipole expansion of the sound field, it can be known that the sound pressure at the spatial position (x h , y h , z h ) can be expressed as:

[0017]

[0018] where i represents the imaginary unit, k = 2π / λ represents the wave number, λ is the signal wavelength, and for simplicity of representation, let D x = sinθcosφ, D y = sinθsinφ, and D z= cosθ, where α, β, and γ represent the orders of spatial gradients in the x, y, and z directions respectively. The trigonometric function part that does not change with frequency is defined as the multipole mode, i.e.,

[0019] Ψ α,β,γ = (D x ) α (D y ) β (D z ) γ

[0020] That is, the multipole mode is included in each order of spatial gradient. Similarly, according to the relationship between sound pressure and particle velocity, the particle velocity at (x h , y h , z h ) can be expressed as:

[0021]

[0022] where ρ represents the medium density, c represents the sound speed, and μ = x, y, z represent the three particle velocity components of the vector hydrophone. Since for any element z h = 0 in the acoustic vector planar array, the signal received by the velocity sensor in the acoustic vector planar array can be expressed as:

[0023]

[0024] It can be seen from the above formula that the acoustic vector planar array contains the multipole mode D μ Ψ α,β,0 . Among them, if the particle velocity component is μ = x, the multipole mode included in the particle velocity gradient of the α + β + 0 order can be expressed as D μ Ψ α,β,0 = Ψ α+1,β,0 . Similarly, when μ = y, z, the expression of D μ Ψ α,β,0 can be obtained. Therefore, the acoustic vector planar array only contains the multipole modes of γ = 0 (μ ≠ z) and γ = 1 (μ = z).

[0025] Furthermore, in step 3,

[0026] For the spherical harmonic function Y n m (θ, φ) with m ≥ 0 and the multipole mode Ψ α,β,γ with γ ≤ 1, the relationship is expressed as:

[0027]

[0028] where, is 's constant coefficient, represents the binomial coefficient, Denotes rounding down; for simplicity of representation, the relationship between the spherical harmonic function and the multipole mode in the above formula is expressed in matrix form:

[0029]

[0030] where is a vector composed of each multipole mode coefficient, and M D is a vector composed of all multipole modes with α + β + γ ≤ N and γ ≤ 1. For the spherical harmonic function in the part where m < 0, according to the conjugate property of the spherical harmonic function, we can obtain That is Therefore, the spherical harmonic function with m < 0 does not require additional multipole modes; in summary, for spherical harmonic functions of any order it can be represented only by multipole modes with α + β + γ ≤ n and γ ≤ 1, which is consistent with the fact that the acoustic vector planar array only contains multipole modes with γ ≤ 1; representing all spherical harmonic functions of order n ≤ N by multipole modes and writing them in matrix form, we have:

[0031] b Y (θ, φ) = EM D

[0032] where E is the coefficient matrix composed of multipole mode coefficients when representing each order of spherical harmonic function by multipole modes, and can be specifically expressed as

[0033] Furthermore, in step 4,

[0034] Construct a small-size planar array based on acoustic vector hydrophones, and assume that the amplitudes of the particle velocity components have been compensated to unity, and establish its received signal model:

[0035]

[0036] where The operation represents the Kronecker product, The elements in the left vector respectively represent the directivities of the acoustic pressure p, x, y, and z velocity channels of the array element; a(θ, φ) = [a 1 (θ, φ), a 2 (θ, φ),..., a M (θ, φ)] T represents the steering vector of the array acoustic pressure channel, M represents the number of array elements, where a h (θ, φ) = exp[ik(x h sinθcosφ + y h sinθsinφ + z hcosθ); h = 1, 2, ..., M;

[0037] For α + β + γ = 0, 1, the multipole mode Ψ α,β,γ is respectively included in the 0th - order sound - pressure gradient and the 0th - order particle - velocity gradient, which means that Ψ α,β,γ can be directly obtained by the acoustic - vector hydrophone at the coordinate origin; The relationship between Ψ α,β,γ and the array steering vector A(θ, φ) can be expressed as:

[0038] Ψ α,β,γ = r α,β,γ T A(θ, φ)

[0039] where r α,β,γ is the coefficient vector composed of the coefficients of each receiving sensor of the array when obtaining the multipole mode Ψ α,β,γ ; The h - th, (h + M) - th, (h + 2M) - th, and (h + 3M) - th elements in r α,β,γ respectively represent the coefficients of the sound pressure p, and the particle - velocity components in the x, y, and z directions of the h - th acoustic - vector hydrophone when extracting the multipole mode Ψ α,β,γ ;

[0040] For the multipole mode Ψ in the case of α + β + γ≥2 α,β,γ = D μ Ψ α′,β′,γ′ Performing a difference - approximation operation on each velocity channel of the array to obtain the high - order particle - velocity gradient:

[0041]

[0042] where d represents the adjacent - element spacing, is the difference - approximation particle - velocity gradient when, it is a column vector composed of the binomial coefficients of each receiving sensor, ε α′,β′,γ′ is the constant coefficient introduced in the (α′ + β′ + γ′) - th order difference approximation, and the relationship between the multipole mode and the array steering vector:

[0043]

[0044] where F α′+β′+0 (k)=1 / (ikd) α′+β′+0 is the (α′ + β′ + 0) - th order difference - amplitude compensation factor when kd << 1; The coefficients of each receiving channel when obtaining the multipole mode with α + β + γ≥2 and γ≤1 The vector M D composed of each order of multipole modes and the relationship between the array steering vector A(θ, φ) can be expressed as:

[0045] M D= RA(θ, φ)

[0046] where R is an (N + 1) α,β,γ × 4M matrix R = [r 2 , r 0,0,0 ,..., r 1,0,0 ...] α,β,γ ...] T , so b Y (θ, φ) = ERA(θ, φ).

[0047] Furthermore, in step 5,

[0048] To solve the beamforming weight coefficients, the beam pattern is written in the following form:

[0049] B N (θ, φ) = w(θ d , φ d ) H A(θ, φ) = c T ERA(θ, φ)

[0050] where w(θ d , φ d ) represents the beamforming weight vector with the steering angle of (θ d , φ d ), and the superscript H represents the conjugate transpose; the weight vector of the beamformer can be obtained by combining the above coefficients as:

[0051] w(θ d , φ d ) H = c T ER

[0052] Using the above weight vector to weight the small-size vector planar array can obtain the pre-designed three-dimensional beam pattern and achieve three-dimensional beamforming.

[0053] The present invention proposes an electronic device, including a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, the steps of the three-dimensional beamforming method based on a small-size planar array of acoustic vector hydrophones are implemented.

[0054] The present invention also proposes a computer-readable storage medium for storing computer instructions, and when the computer instructions are executed by a processor, the steps of the three-dimensional beamforming method based on a small-size planar array of acoustic vector hydrophones are implemented.

[0055] The present invention realizes three-dimensional beamforming by using a planar array, avoids the vertical direction amplitude-phase error introduced under the influence of deep-sea waveguide, and solves the problem that it is difficult for existing low-frequency small-size arrays to be applied in deep-sea three-dimensional space detection. Description of the Drawings

[0056] Figure 1 is the flowchart of the method of the present invention;

[0057] Figure 2 is the third-order three-dimensional beam pattern pointing to the angle (45°, 45°);

[0058] Figure 3 is the geometric schematic diagram of a 9-element vector planar array;

[0059] Figure 4 is the schematic diagram of the differential approximation of the second-order vibration velocity spatial gradient;

[0060] Figure 5 is the cross-sectional view of the third-order beam pattern at the azimuth angle of 0° and the elevation angle of 90° under different d / λ conditions; (a) Cross-sectional view at the azimuth angle of 0°; (b) Cross-sectional view at the elevation angle of 90°;

[0061] Figure 6 is the variation diagram of the array gain and the white noise gain with d / λ under different orders; (a) Array gain; (b) White noise gain. Detailed Embodiment

[0062] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0063] Combined with Figures 1 - 6 , the present invention proposes a three-dimensional beamforming method for a small-sized planar array based on acoustic vector hydrophones, which specifically includes the following steps:

[0064] Step 1: Design the required three-dimensional beam pattern, expand the designed three-dimensional beam pattern with the complete orthogonal basis functions (spherical harmonic functions) in the spherical coordinate system, and determine the coefficients of each order of spherical harmonic functions;

[0065] Step 2: Determine the multipole modes included in the acoustic vector planar array;

[0066] Step 3: Further expand each order of spherical harmonic functions into the multipole modes owned by the acoustic vector planar array, and determine the coefficients of each multipole mode;

[0067] Step 4: Use the finite difference technique to extract the required multipole modes, design the required planar array, and determine the coefficients of each receiving sensor when extracting the multipole modes;

[0068] Step 5: Combine the obtained coefficients to get the required beamformer weight vector and obtain the designed 3D beam pattern.

[0069] In step 1,

[0070] Construct a spherical coordinate system with the array center as the origin, and design the required 3D beam pattern B(θ,φ), where θ and φ represent the elevation angle and azimuth angle respectively, and expand it using spherical harmonic functions Y n m (θ,φ):

[0071]

[0072] Since only a finite number of spherical harmonic functions can be used in practical applications, the infinite series in the above formula is truncated to the Nth order, that is, N represents the highest order of the spherical harmonic function, where c n,m represents the coefficient after the beam pattern expansion, c = [c , c 0,0 ,..., c -1,1 ,..., c N,N ) T is an (N + 1) n,m ×1 vector, and the superscript T represents the matrix transpose. 2 is an (N + 1) ×1 column vector composed of spherical harmonic functions with n ≤ N and |m| ≤ n. 2

[0073] In step 2,

[0074] From the multipole expansion of the sound field, it can be known that the sound pressure at the spatial position (x h , y h , z h ) can be expressed as:

[0075]

[0076] where i represents the imaginary unit, k = 2π / λ represents the wave number, and λ is the signal wavelength. For simplicity of representation, let D x = sinθcosφ, D y = sinθsinφ, and D z = cosθ respectively. α, β, and γ represent the spatial gradient orders in the x, y, and z directions respectively. Define the trigonometric function part that does not change with frequency as the multipole mode, that is

[0077] Ψ α,β,γ = (D x ) α (D y ) β (Dz ) γ

[0078] That is, the multipole modes are included in the spatial gradients of each order; similarly, according to the relationship between the sound pressure and the particle velocity, the particle velocity at (x h , y h , z h ) can be expressed as:

[0079]

[0080] where ρ represents the medium density, c represents the sound speed, μ = x, y, z represent the three particle velocity components of the vector hydrophone. Since for any element z h = 0 in the acoustic vector planar array, the received signal of the velocity sensor in the acoustic vector planar array can be expressed as:

[0081]

[0082] It can be seen from the above formula that the acoustic vector planar array contains the multipole mode D μ Ψ α,β,0 , where if the particle velocity component with μ = x, the multipole mode contained in the particle velocity gradient of the α + β + 0 order can be expressed as D μ Ψ α,β,0 = Ψ α+1,β,0 . Similarly, when μ = y, z, the expression of D μ Ψ α,β,0 can be obtained. Therefore, the acoustic vector planar array only contains the multipole modes with γ = 0 (μ ≠ z) and γ = 1 (μ = z).

[0083] In step 3,

[0084] For the spherical harmonic function Y n m (θ, φ) with m ≥ 0 and the multipole mode Ψ α,β,γ with γ ≤ 1, the relationship is expressed as:

[0085]

[0086] where, is 's constant coefficient, represents the binomial coefficient, represents the floor function; for the sake of simplicity, the relationship between the spherical harmonic function and the multipole mode in the above formula is expressed in matrix form:

[0087]

[0088] where is a vector composed of each multipole mode coefficient, M D is a vector composed of all multipole modes with α + β + γ ≤ N and γ ≤ 1. For the spherical harmonic functions with m < 0, according to the conjugate property of spherical harmonic functions, we can get That is Therefore, no additional multipole modes are required for the spherical harmonic functions with m < 0; In summary, for the spherical harmonic functions Y of any order n m (θ, φ) can be represented only by the multipole modes with α + β + γ ≤ n and γ ≤ 1, which is consistent with the fact that the acoustic vector planar array only contains the multipole modes with γ ≤ 1; Represent all spherical harmonic functions of order n ≤ N in terms of multipole modes and write them in matrix form as:

[0089] b Y (θ, φ) = EM D

[0090] where E is the coefficient matrix composed of multipole mode coefficients when representing each order of spherical harmonic functions in terms of multipole modes, and can be specifically expressed as

[0091] In step 4

[0092] Construct a small-size planar array based on acoustic vector hydrophones, and assume that the amplitudes of the particle velocity components have been compensated to unity, and establish its received signal model:

[0093]

[0094] where The operation represents the Kronecker product The elements in the left vector respectively represent the directivities of the acoustic pressure p, x, y, and z velocity channels of the array element; a(θ, φ) = [a 1 (θ, φ), a 2 (θ, φ),..., a M (θ, φ)] T represents the steering vector of the array acoustic pressure channel, M represents the number of array elements, where a h (θ, φ) = exp[ik(x h sinθcosφ + y h sinθsinφ + z h cosθ)]; h = 1, 2,..., M;

[0095] For α + β + γ = 0, 1, the multipole mode Ψ α,β,γ is respectively included in the 0th-order acoustic pressure gradient and the 0th-order velocity gradient, which means that Ψ α,β,γ can be directly obtained through the acoustic vector hydrophone at the coordinate origin; Ψ α,β,γThe relationship with the array steering vector A(θ, φ) can be expressed as:

[0096] Ψ α,β,γ = r α,β,γ T A(θ, φ)

[0097] where r α,β,γ is the coefficient vector composed of the coefficients of each receiving sensor of the array when obtaining the multipole mode Ψ α,β,γ ; the h, (h + M), (h + 2M), and (h + 3M) elements in r α,β,γ respectively represent the coefficients of the sound pressure p, and the particle velocity components in the x, y, and z directions of the h-th acoustic vector hydrophone when extracting the multipole mode Ψ α,β,γ ;

[0098] For the multipole mode Ψ in the case of α + β + γ ≥ 2 α,β,γ = D μ Ψ α′,β′,γ′ , perform a differential approximation operation on each vibration velocity channel of the array to obtain the high-order vibration velocity gradient:

[0099]

[0100] where d represents the adjacent element spacing, is the differential approximation vibration velocity gradient when, the column vector composed of the binomial coefficients of each receiving sensor, ε α′,β′,γ′ is the constant coefficient introduced in the α' + β' + γ'-th order differential approximation, and can be determined according to the relationship between the differential element spacing in a certain direction and d when performing the differential approximation in that direction. The relationship between the multipole mode and the array steering vector:

[0101]

[0102] where F α′+β′+0 (k) = 1 / (ikd) α′+β′+0 is the α' + β' + 0-th order differential amplitude compensation factor when kd << 1; the coefficients of each receiving channel when obtaining the multipole mode with α + β + γ ≥ 2 and γ ≤ 1 The vector M D composed of each order of multipole mode and the array steering vector A(θ, φ) can be expressed as:

[0103] M D = RA(θ, φ)

[0104] where R is the (N + 1) α,β,γ × 4M matrix composed of r 2 R = [r 0,0,0 , r 1,0,0 ,..., rα,β,γ ...] T , so b Y (θ, φ) = ERA(θ, φ).

[0105] In step 5,

[0106] To solve the beamforming weight coefficients, the beam pattern is written in the following form:

[0107] B N (θ, φ) = w(θ d , φ d ) H A(θ, φ) = c T ERA(θ, φ)

[0108] where w(θ d , φ d ) represents the beamformer weight vector with the steering angle (θ d , φ d ); the superscript H represents the conjugate transpose; the weight vector of the beamformer can be obtained by combining the above coefficients as follows:

[0109] w(θ d , φ d ) H = c T ER

[0110] Using the above weight vector to weight the small - size vector planar array can obtain the pre - designed three - dimensional beam pattern and achieve three - dimensional beamforming.

[0111] The present invention proposes an electronic device, including a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, the steps of the three - dimensional beamforming method for a small - size planar array based on acoustic vector hydrophones are implemented.

[0112] The present invention also proposes a computer - readable storage medium for storing computer instructions. When the computer instructions are executed by the processor, the steps of the three - dimensional beamforming method for a small - size planar array based on acoustic vector hydrophones are implemented.

[0113] Embodiment

[0114] Design the required three - dimensional beam pattern. Taking the maximum directivity factor as the criterion, design a three - dimensional beam pattern with the maximum spherical harmonic function order N = 3 and the steering angle (θ d , φ d ). As shown in Figure 2 is the designed three - dimensional beam pattern with (θ d , φ d ) = (45°, 45°).

[0115]

[0116] That is, after the designed three-dimensional beam pattern is expanded by spherical harmonic functions, the coefficient vector composed of the coefficients of each order of spherical harmonic functions

[0117] According to the connection between spherical harmonic functions and multipole modes, each order of spherical harmonic functions is expanded into the form of multipole modes. For the spherical harmonic function Y n m(θ, φ) and the multipole mode Ψ α,β,γ =(sinθcosφ) α (sinθsinφ) β (cosθ) γ The connection between them can be expressed as:

[0118]

[0119] where is a constant coefficient of represents the binomial coefficient, represents rounding down. For the sake of simplicity, the relationship between the spherical harmonic function and the multipole mode in the above formula is expressed in matrix form:

[0120]

[0121] where M D is the vector composed of all multipole modes with α + β + γ ≤ N and γ ≤ 1. For example, if N = 2, then M D contains all multipole modes M with α + β + γ = 0, 1, 2 and γ = 0, 1 D =[Ψ 0,0,0 , Ψ 1,0,0 , Ψ 0,1,0 , Ψ 0,0,1 , Ψ 2,0,0 , Ψ 1,1,0 , Ψ 1,0,1 , Ψ 0,2,0 , Ψ 0,1,1 T . is the (N + 1) D ×1 column vector composed of the coefficients of each order of multipole modes when expressing Y n m (θ, φ), and the specific value of each element can be calculated according to the above formula. Taking 2 as an example: For example:

[0122] ​

[0123] 0 p×q For the spherical harmonics of the all-zero matrix of p×q for the part where m < 0, according to the conjugate property of the spherical harmonics, it can be obtained that That is All spherical harmonics with order n ≤ N Are represented by multipole modes and written in matrix form as:

[0124] b Y (θ, φ) = EM D

[0125] Where E is the coefficient matrix composed of the multipole mode coefficients when representing each order of spherical harmonics with multipole modes, and specifically can be expressed as

[0126] The maximum order of spherical harmonics N = 3. Therefore, the maximum order of the required multipole modes is α + β + γ = 3, and γ ≤ 1. For this reason, a 3×3 uniform rectangular vector array as shown Figure 3 Is used to extract the required multipole modes subsequently. It includes array elements numbered 1 to 9. The array element numbered 5 at the center of the array is also the reference array element, and the reference sound pressure response p 0 = p 5 . The distance between any two adjacent array elements along the coordinate axes is d. The steering vector form of the received signal is:

[0127]

[0128] The operation represents the Kronecker product, The elements in the left vector respectively represent the directivity of the array element sound pressure p, x, y, and z vibration velocity channels; a(θ, φ) = [a 1 (θ, φ), a 2 (θ, φ),..., a 9 (θ, φ)] T Represents the steering vector of the array sound pressure channel, where a h (θ, φ) = exp[ik(x h sinθcosφ + y h sinθsinφ + z h cosθ)], h = 1, 2,..., 9;

[0129] For α + β + γ = 0, 1, the multipole modes Ψ α,β,γ Are respectively included in the 0th-order sound pressure gradient and the 0th-order vibration velocity gradient, which means that Ψ α,β,γ Can be directly obtained through the acoustic vector hydrophone at the origin of coordinates. The connection between Ψ α,β,γ And the array steering vector A(θ, φ) can be expressed as:

[0130] Ψ α,β,γ = r α,β,γ T A(θ, φ)

[0131] where r α,β,γ is the coefficient vector composed of the coefficients of each receiving sensor in the array when obtaining the multipole mode Ψ α,β,γ . The h, (h + M), (h + 2M), and (h + 3M) elements in r α,β,γ respectively represent the coefficients of the sound pressure p, and the particle velocity components in the x, y, and z directions of the h-th acoustic vector hydrophone when extracting the multipole mode Ψ α,β,γ .

[0132] For the multipole mode Ψ in the case of α + β + γ ≥ 2 α,β,γ = D μ Ψ α′,β′,γ′ Differential approximation operations are performed on each receiving channel of the array to obtain the high-order partial derivatives of the sound field:

[0133]

[0134] where is the differential approximation velocity gradient when, the column vector composed of the binomial coefficients of each receiving sensor, ε α′,β′,γ′ is the constant coefficient introduced in the α′ + β′ + γ′-order differential approximation, which can be determined according to the relationship between the differential element spacing in a certain direction and d when performing differential approximation in that direction. To illustrate in detail and ε α′,β′,γ′ , taking all the second-order velocity gradients that can be extracted by the Figure 3 vector array as an example, the coefficients of each element during differential approximation are obtained as shown in Figure 4 . For the convenience of understanding, the top view of the array (i.e., the xoy plane) is selected for illustration. In the figure, red, blue, and green represent the velocity sensors in the x, y, and z directions respectively. The red numbers in the upper left corner of the element represent the coefficients of the sensor during differential approximation. The solid line of the element border represents a positive coefficient, and the dashed line represents a negative coefficient. The numbers with the '#' sign below the element represent the element numbers. The connection between the multipole mode and the array steering vector:

[0135]

[0136] where F α′+β′+0 (k) = 1 / (ikd) α′+β′+0 is the α′ + β′ + 0-order differential amplitude compensation factor when kd << 1. The coefficients of each receiving channel when obtaining the multipole mode with α + β + γ ≥ 2 and γ ≤ 1 Taking Figure 4 (e) as an example: ​

[0137]

[0138] The vector M composed of the required multipole modes of each order D The relationship with the array manifold A(θ, φ) can be expressed as:

[0139] M D = RA(θ, φ)

[0140] where R is composed of r α,β,γ The (N + 1) 2 ×4M matrix R = [r 0,0,0 , r 1,0,0 ,..., r α,β,γ ...] T , so b Y (θ, φ) = ERA(θ, φ).

[0141] According to the relationship between the beam pattern and the beamformer, the weighted vector of the beamformer is obtained:

[0142] B N (θ, φ) = w H (θ d , φ d )A(θ, φ) = c T ERA(θ, φ)

[0143] The weighted vector of the beamformer is obtained as:

[0144] w H (θ d , φ d ) = c T ER

[0145] Using the above weighted vector to perform weighted processing on the small-size vector planar array can obtain the pre-designed 3D beam pattern and achieve 3D beamforming.

[0146] To verify the applicable range of the proposed method, taking the highest-order spherical harmonic N = 3 that the array can extract as an example, with the steering angle being (90°, 180°), the cross-sectional views of the beam pattern at the azimuth angle of 0° and the elevation angle of 90° are respectively analyzed with the change of d / λ, as shown in Figure 5 (a) and 5(b). In the case of 0.01 ≤ d / λ ≤ 0.2 in the present invention, the main lobe width of the beam pattern basically remains unchanged. Although the side lobes gradually increase, the error from the ideal beam pattern is small. As d / λ continues to increase, since the error introduced by the differential approximation also gradually increases at this time, when 0.3 ≤ d / λ ≤ 0.5, the differential approximation error is too large, the beam pattern is severely distorted, the main lobe of the azimuth angle has deviated from the correct position, and the beamforming algorithm fails. It can be seen from this that the present invention is applicable to low-frequency signal processing.

[0147] Finally, the gain and robustness of the present invention are verified to Figure 3 obtain the array gain and white noise gain of the beamforming method of the present invention when N = 1, 2, and 3 respectively for the shown arrays. The variation trend of the array gain with d / λ is as Figure 6 (a) shown, where the black dashed line is the theoretical array gain of the corresponding order, and the red solid line represents the variation of the simulated array gain with d / λ. It can be seen from the simulation results that the small-sized vector planar array and its beamforming method established by the present invention have a relatively high array gain. Taking N = 3 and d / λ = 0.1 as an example, the array gain can reach 12 dB at this time. Under the condition of d / λ ≤ 0.2, the array gain basically coincides with the corresponding dashed line and remains unchanged, which means that this method can obtain high gain in a relatively wide low-frequency band. However, from Figure 6 (b), it can be seen that as the frequency decreases, the WNG will decrease rapidly. At this time, the array will be more sensitive to the amplitude-phase errors and self-noise of the elements, and the robustness of the entire system will also be worse. At the same time, by comparing the array gain and white noise gain in the case of different orders N, it can be known that as the maximum spherical harmonic order N increases, the array gain gradually increases and the white noise gain gradually decreases. In practical applications, a trade-off should be made between the two according to the requirements and actual conditions.

[0148] The above has introduced in detail a three-dimensional beamforming method, device, and medium of a small-sized planar array based on acoustic vector hydrophones proposed by the present invention. Specific examples are used in this article to elaborate on the principle and implementation manner of the present invention. The description of the above embodiments is only used to help understand the method and its core idea of the present invention; at the same time, for those of ordinary skill in the art, according to the idea of the present invention, there will be changes in the specific implementation manner and application scope. In summary, the content of this specification should not be construed as a limitation to the present invention.

Claims

1. A three-dimensional beamforming method for a small-sized planar array based on a vector hydrophone, characterized in that, it specifically includes the following steps: Step 1: Design the required three-dimensional beam pattern, expand the designed three-dimensional beam pattern using spherical harmonic functions in the spherical coordinate system, and determine the coefficients of each order of spherical harmonic functions; Step 2: Determine the multipole modes included in the vector hydrophone planar array; In step 2, let D x = sinθcosφ, D y = sinθsinφ, and D z = cosθ. α, β, and γ represent the spatial gradient orders in the x, y, and z directions respectively. The trigonometric function part that does not change with frequency is defined as the multipole mode, i.e., Ψ α,β,γ = (D x ) α (D y ) β (D z ) γ that is, the multipole modes are included in the spatial gradients of each order; Step 3: Further expand each order of spherical harmonic functions into the multipole modes possessed by the vector hydrophone planar array, and determine the coefficients of each multipole mode; In Step 3, the relationship between the spherical harmonic function and the multipole mode is expressed in matrix form: Among them is a vector composed of each multipole mode coefficient, M D is a vector composed of all multipole modes with α + β + γ ≤ N and γ ≤ 1. For the spherical harmonic functions with m < 0, according to the conjugate property of the spherical harmonic functions, we can get That is Therefore, no additional multipole modes are required for the spherical harmonic functions with m < 0; in summary, for spherical harmonic functions of any order can be represented only by multipole modes with α + β + γ ≤ n and γ ≤ 1, which is consistent with the fact that the acoustic vector planar array only contains multipole modes with γ ≤ 1; representing all spherical harmonic functions of order n ≤ N by multipole modes and writing them in matrix form, we have: b Y (θ, φ) = EM D where E is a coefficient matrix composed of multipole mode coefficients when representing spherical harmonic functions of each order in multipole modes, and can be specifically expressed as Step 4: Use the finite difference technique to extract the required multipole modes, design the required planar array, and determine the coefficients of each receiving sensor when extracting the multipole modes; Step 5: Synthesize the obtained coefficients to obtain the weighted vector of the required beamformer, and obtain the designed three-dimensional beam pattern.

2. The method according to claim 1, characterized in that, in Step 1, Construct a spherical coordinate system with the center of the array as the origin, design the required three-dimensional beam pattern B(θ, φ), where θ and φ represent the elevation angle and azimuth angle respectively, and expand it using spherical harmonic functions of each order Expand: In the above formula, the infinite series after expansion is truncated to the Nth order, where N represents the highest order of the spherical harmonic function, and c n,m represents the coefficient after the beam pattern is expanded . c = [c 0,0 , c -1,1 ,..., c N,N T is an (N + 1) n,m × 1 vector composed of c 2 coefficients. The superscript T represents the matrix transpose, is an (N n m + 1) + × 1 column vector composed of spherical harmonic functions Y 2 (θ, φ) with n ≤ N and |m| ≤ n​ 3. The method according to claim 2, characterized in that, in Step 2, From the multipole expansion of the sound field, it can be known that the sound pressure at the spatial position (x h , y h , z h ) is expressed as: where \(i\) represents the imaginary unit, \(k = \frac{2\pi}{\lambda}\) represents the wavenumber, and \(\lambda\) is the signal wavelength; according to the relationship between sound pressure and particle velocity, the particle velocity at \((x h ,y h ,z h ) is expressed as: Among them, ρ represents the medium density, c represents the sound speed, μ = x, y, z represents the three particle vibration velocity components of the vector hydrophone. Since for any element z in the acoustic vector planar array h = 0, the signal received by the vibration velocity sensor in the acoustic vector planar array is expressed as: It can be seen from the above formula that the acoustic vector planar array contains the multipole mode D μ Ψ α,β,0 , where if the particle velocity component is μ = x, the multipole mode included in the particle velocity gradient of the α + β + 0 order can be expressed as D μ Ψ α,β,0 = Ψ α+1,β,0 . Similarly, when μ = y, z, the expression of D μ Ψ α,β,0 can be obtained. Therefore, the acoustic vector planar array only contains the multipole modes of γ = 0 (μ ≠ z) and γ = 1 (μ = z).

4. The method according to claim 3, characterized in that, in Step 3, For spherical harmonics with m ≥ 0 and multipole modes Ψ with γ ≤ 1 α,β,γ the connection is expressed as: Among them, is a constant coefficient of, represents the binomial coefficient, represents rounding down.

5. The method according to claim 4, characterized in that, in Step 4, construct a small-sized planar array based on a vector hydrophone, and assume that the amplitude of the particle velocity component has been compensated to unity, and establish its received signal model: Among them The operation represents the Kronecker product The elements in the left vector respectively represent the array element sound pressure p, the directivities of the x, y, and z vibration velocity channels; a(θ, φ) = [a 1 (θ, φ), a 2 (θ, φ),..., a M (θ, φ)] T represents the steering vector of the array sound pressure channel, M represents the number of array elements, where a h (θ, φ) = exp[ik(x h sinθcosφ + y h sinθsinφ + z h cosθ)]; h = 1, 2,..., M; For α + β + γ = 0, 1, the multipole mode Ψ α,β,γ is respectively included in the 0th - order sound - pressure gradient and the 0th - order particle - velocity gradient, which means that Ψ α,β,γ can be directly obtained by a vector - hydrophone at the coordinate origin; the relationship between Ψ α,β,γ and the array steering vector A(θ, φ) can be expressed as: Ψ α,β , γ = r α,β,γ T A(θ, φ) where r α,β,γ is the coefficient vector composed of the coefficients of each receiving sensor in the array when obtaining the multipole mode Ψ α,β,γ ; the h-th, (h + M)-th, (h + 2M)-th, and (h + 3M)-th elements in r α,β,γ respectively represent the coefficients of the sound pressure p, the particle velocity components in the x, y, and z directions of the h-th acoustic vector hydrophone when extracting the multipole mode Ψ α,β,γ ; For the multipole mode Ψ in the case of α + β + γ ≥ 2 α,β,γ = D μ Ψ α′,β′,γ′ , perform a differential approximation operation on each vibration velocity channel of the array to obtain a high-order vibration velocity gradient: where d represents the adjacent element spacing, is the differential approximate vibration velocity gradient When it is, the column vector composed of the binomial coefficients of each receiving sensor, ε α′,β′,γ′ is the constant coefficient introduced in the α′+β′+γ′-order differential approximation. The relationship between the multipole mode and the array steering vector: where F α′+β′+0 (k) = 1 / (ikd) α′+β′+0 is the differential amplitude compensation factor of the α′+β′+0th order when kd << 1; obtain the coefficients of each receiving channel when obtaining the multipole mode with α+β+γ≥2 and γ≤1 the vector M composed of each order of multipole mode D The relationship with the array steering vector A(θ,φ) can be expressed as: M D = RA(θ, φ) where R is an (N + 1) α,β,γ × 4M matrix R = [r 2 , r 0,0,0 ,..., r 1,0,0 ...] α,β,γ ...] T , so b Y (θ, φ) = ERA(θ, φ).

6. The method according to claim 5, characterized in that, in Step 5, for solving the beamforming weighting coefficients, the beam pattern is written in the following form: B N (θ, φ) = w(θ d , φ d ) H A(θ, φ) = c T ERA(θ, φ) where \(w(\theta\) d ,\(\varphi\) d ) represents the beamformer weight vector with the steering angles of \((\theta\) d ,\(\varphi\) d ). The superscript \(H\) represents the conjugate transpose. The weight vector of the beamformer can be obtained by combining the above coefficients as follows: w(θ d ,φ d ) H =c T ER Using the above weighted vector to perform weighted processing on the small-sized vector planar array can obtain the pre-designed three-dimensional beam pattern, and realize three-dimensional beamforming.

7. An electronic device, including a memory and a processor, the memory stores a computer program, characterized in that, when the processor executes the computer program, it implements the steps of the method according to claims 1-6.

8. A computer-readable storage medium for storing computer instructions, characterized in that, when the computer instructions are executed by the processor, they implement the steps of the method according to claims 1-6.

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