Multifrequency synchronous two-dimensional netless compressive beamforming method

Abstract

The application discloses a multi-frequency synchronous two-dimensional net-free compression beam forming method, which comprises the following steps: 1, two-dimensional continuous multi-frequency synchronous model construction; 2, establishment of an atomic norm minimization problem and an equivalent dual problem; 3, solution of the dual problem; and 4, sound source DOA and intensity estimation. The technical effect of the application is that the spatial aliasing problem borne by the existing single-frequency processing method is effectively inhibited, and a sound source with a wavelength less than twice the minimum distance between microphones can be accurately identified.

Description

Technical Field

[0001] This invention relates to the field of sound source identification technology, specifically a multi-frequency synchronous two-dimensional meshless compressed beamforming method. Background Technology

[0002] Two-dimensional meshless compressed beamforming technology based on rectangular planar microphone array measurement has gained significant attention in engineering fields such as aero-engine noise detection, automotive aerodynamic and vibration noise tracing, and wind turbine blade noise identification due to its outstanding advantages such as wide sound source identification area (covering the hemisphere in front of the array), high positioning accuracy (effectively avoiding basis function mismatch problems), and robust overall performance. It has become one of the important means of acoustic measurement and sound source identification.

[0003] However, this technology still faces a key challenge in practical applications: it needs to be matched with regular arrays such as uniform rectangular arrays and sparse rectangular arrays. When the number of microphones and the equivalent diameter of the array are fixed, regular arrays typically have a larger minimum microphone spacing. Therefore, when measuring high-frequency sound sources, they are more susceptible to spatial aliasing, making it difficult to correctly identify the sound source.

[0004] Specifically, when the wavelength of a sound wave is less than twice the minimum spacing between microphones in the array, aliasing occurs in the wavenumber domain, causing false or mirror sources to appear in the sound source localization results, severely interfering with the judgment of the sound source's location and intensity. Although beamforming methods based on single-frequency processing have shown good performance within a certain frequency band, they cannot fundamentally overcome the spatial sampling limitations determined by the array geometry, thus limiting their ability to identify high-frequency sound sources.

[0005] This bottleneck limits the further application and promotion of this technology in high-frequency noise detection scenarios. In particular, in modern industrial and scientific research fields that require wide-bandwidth and high-resolution sound source identification, how to effectively suppress spatial aliasing and expand the identifiable frequency band without significantly increasing the number of microphones has become a key problem that urgently needs to be solved.

[0006] In summary, although existing two-dimensional meshless compressed beamforming methods based on uniform rectangular planar arrays have many advantages, the spatial aliasing problem restricts their reliable application across the entire frequency band, especially the high-frequency band. Summary of the Invention

[0007] The purpose of this invention is to provide a multi-frequency synchronous two-dimensional meshless compressed beamforming method, comprising the following steps:

[0008] Step 1) Establish a two-dimensional multi-frequency synchronization model for the microphone array to acquire signals;

[0009] Step 2) Based on the two-dimensional multi-frequency synchronization model, establish the atomic norm minimization problem and transform it into an equivalent dual problem;

[0010] Step 3) Transform the equivalent dual problem into a semidefinite programming problem and solve it. Construct a spatial spectral function based on the solution of the semidefinite programming problem. .

[0011] Step 4) Based on Estimate the DOA of the sound source and quantize its intensity.

[0012] Furthermore, the microphone array is a uniform rectangular array.

[0013] Furthermore, in step 1), the steps for establishing a two-dimensional multi-frequency synchronization model of the microphone array acquisition signal include:

[0014] Step 1.1) Construct a continuous representation of the multi-frequency sound source signal ,Right now:

[0015] (1)

[0016] in, , , express DOA of the sound source and These represent the elevation angle and the azimuth angle, respectively. , They represent in shaft and The distance between two adjacent microphones along the axial direction. For the Dirac function, Indicates each frequency The row vector formed by the sound pressure generated by the sound source at the origin of the coordinate system. Indicates the sound source index. Indicates the total number of sound sources. Indicates frequency index, Indicates the total number of frequencies;

[0017] Step 1.2) Construct sound sources at various frequencies in The sound pressure vector at the microphone ,Right now:

[0018] (2)

[0019] in, Indicates the index of each microphone. ,symbol" " represents the Hadamard product, For the speed of sound, It is a row vector composed of the center frequencies of discrete multi-frequency sub-bands. , This indicates the formation of a diagonal matrix with the vectors inside the brackets as its diagonal lines;

[0020] Step 1.3) Construct the matrix ,Right now:

[0021] (3)

[0022] In the formula, the symbol " " represents the Khatri-Rao product;

[0023] Step 1.4) Let Construct the sound pressure expression, that is:

[0024] (4)

[0025] in, , Sound pressure ;

[0026] Step 1.5) Introduce noise interference into the sound pressure expression to construct a sound pressure measurement module. The expression is used as a two-dimensional multi-frequency synchronization model for the signals acquired by the microphone array;

[0027] Measuring sound pressure The expression is as follows:

[0028] (5)

[0029] In the formula, This is due to noise interference.

[0030] Furthermore, the origin of the coordinate system is... The location of the microphone.

[0031] Furthermore, in step 2), the atomic norm minimization problem is as follows:

[0032] (6)

[0033] In the formula, atomic norm As shown below:

[0034] (7)

[0035] Represents the trace of a matrix, " denotes the Frobenius norm; This is the upper bound of the noise Frobenius norm.

[0036] Furthermore, in step 2), the equivalent dual problem is as follows:

[0037] (8)

[0038] In the formula, for The List, It is a Lagrange multiplier.

[0039] Furthermore, in step 2), the steps to transform the atomic norm minimization problem into an equivalent dual problem include:

[0040] Step 2.1) Construct the Lagrangian function for the atomic norm minimization problem. ,Right now:

[0041] (9)

[0042] in, and For Lagrange multipliers, This indicates the conjugate transpose. Indicates taking the real part;

[0043] Step 2.2) Convert the Lagrange function The maximum lower bound as the dual function ,Right now:

[0044] (10)

[0045] In the formula, "inf" represents the infimum;

[0046] Step 2.3) Solve ,get:

[0047] (11)

[0048] Step 2.4) Construct the optimal noise matrix ;

[0049] exist Calculate the dual function at point, and at Maximizing the top yields:

[0050] (12)

[0051] Step 2.5) Construct the optimal dual variables ;

[0052] Determine the optimal noise matrix and optimal dual variables The dual function at that point, i.e.:

[0053] (13)

[0054] Step 2.6) Determine The lower bound is used to construct the equivalent dual problem. The steps include:

[0055] Step 2.6.1) Construct the function lower bound expression ;in, This represents the modulo operation on each element of the matrix. for 3D identity matrix for and The element-level angle between them; when hour, Non-negative, lower bound is 0; when At that time, the lower boundary is ;

[0056] Step 2.6.2) Based on function The lower bound expression determines the dual problem, i.e.:

[0057] (14)

[0058] Furthermore, in step 3), the semidefinite programming problem is as follows:

[0059] (15)

[0060]

[0061] in, Let be a Hermitian matrix, satisfying ; Represents the first of the matrix A column vector consisting of diagonal elements. Indicates the diagonal offset. The main diagonal, The time indicates the first one above the main diagonal. Secondary diagonal, The time indicates the first line below the main diagonal. Secondary diagonal; This indicates a summation operation on all elements of the vector or matrix within the parentheses;

[0062] Furthermore, in step 3), the semidefinite programming problem is solved using the SDPT3 solver in the CVX toolbox.

[0063] Furthermore, in step 4), the steps of estimating the DOA of the sound source and quantizing its intensity include:

[0064] Step 4.1) Define the spatial spectral function, i.e.:

[0065] (16)

[0066] In the formula, for The List; For spatial spectral functions;

[0067] Step 4.2) Mid-peak value greater than The number of peaks is denoted as ,Will Each peak position is used as Estimated location of each sound source ;

[0068] Step 4.3) Calculate based on the estimated DOA of the sound source. and And construct a perception matrix ;

[0069] Strength calculated using the least squares method ,Right now:

[0070] (17)

[0071] In the formula, It indicates a false reversal.

[0072] The technical effectiveness of this invention is undeniable. The method provided by this invention first establishes a two-dimensional multi-frequency synchronization model, then constructs an infinite-dimensional atomic norm minimization problem and a finite-dimensional dual problem, and solves them based on CVX. Finally, it uses a spatial spectral function to extract the DOA information and intensity of the sound source from the solution of the dual problem. Due to the introduction of the two-dimensional multi-frequency synchronization model, the method provided by this invention can effectively suppress the spatial aliasing problem suffered by existing single-frequency processing methods when the spatial Nyquist criterion is not satisfied, and accurately identify sound sources with wavelengths less than twice the minimum microphone spacing. Attached Figure Description

[0073] Figure 1 This is the measurement model used in this invention;

[0074] Figure 2 The spatial spectrum function amplitude distribution and sound source identification cloud map of Example 11 at 7000 Hz (aliasing frequency) are shown. Detailed Implementation

[0075] The present invention will be further described below with reference to embodiments, but it should not be construed that the scope of the present invention is limited to the following embodiments. Various substitutions and modifications made based on ordinary technical knowledge and common practices in the art without departing from the above-described technical concept of the present invention should be included within the scope of protection of the present invention.

[0076] Example 1:

[0077] See Figure 1 A multi-frequency synchronous two-dimensional meshless compressed beamforming method includes the following steps:

[0078] Step 1) Establish a two-dimensional multi-frequency synchronization model for the microphone array to acquire signals;

[0079] Step 2) Based on the two-dimensional multi-frequency synchronization model, establish the atomic norm minimization problem and transform it into an equivalent dual problem;

[0080] Step 3) Transform the equivalent dual problem into a semidefinite programming problem and solve it. Construct a spatial spectral function based on the solution of the semidefinite programming problem. .

[0081] Step 4) Based on Estimate the DOA of the sound source and quantize its intensity.

[0082] Example 2:

[0083] The method for multi-frequency synchronous two-dimensional meshless compressed beamforming is the same as in Example 1, except that the microphone array is a uniform rectangular array.

[0084] Example 3:

[0085] The method for multi-frequency synchronous two-dimensional meshless compressed beamforming has the same technical content as any one of embodiments 1-2. Further, in step 1), the step of establishing a two-dimensional multi-frequency synchronous model of the microphone array acquisition signal includes:

[0086] Step 1.1) Construct a continuous representation of the multi-frequency sound source signal ,Right now:

[0087] (1)

[0088] in, , , express DOA of the sound source and These represent the elevation angle and the azimuth angle, respectively. , They represent in shaft and The distance between two adjacent microphones along the axial direction. For the Dirac function, Indicates each frequency The row vector formed by the sound pressure generated by the sound source at the origin of the coordinate system. Indicates the sound source index. Indicates the total number of sound sources. Indicates frequency index, Indicates the total number of frequencies;

[0089] Step 1.2) Construct sound sources at various frequencies in The sound pressure vector at the microphone ,Right now:

[0090] (2)

[0091] in, Indicates the index of each microphone. ,symbol" " represents the Hadamard product, For the speed of sound, It is a row vector composed of the center frequencies of discrete multi-frequency sub-bands. , This indicates the formation of a diagonal matrix with the vectors inside the brackets as its diagonal lines;

[0092] Step 1.3) Construct the matrix ,Right now:

[0093] (3)

[0094] In the formula, the symbol " " represents the Khatri-Rao product;

[0095] Step 1.4) Let Construct the sound pressure expression, that is:

[0096] (4)

[0097] in, , Sound pressure ;

[0098] Step 1.5) Introduce noise interference into the sound pressure expression to construct a sound pressure measurement module. The expression is used as a two-dimensional multi-frequency synchronization model for the signals acquired by the microphone array;

[0099] Measuring sound pressure The expression is as follows:

[0100] (5)

[0101] In the formula, This is due to noise interference.

[0102] Example 4:

[0103] The multi-frequency synchronous two-dimensional meshless compressed beamforming method has the same technical content as any one of embodiments 1-3, further wherein the origin of the coordinate system is... The location of the microphone.

[0104] Example 5:

[0105] The multi-frequency synchronous two-dimensional meshless compressed beamforming method, with the same technical content as any one of embodiments 1-4, further includes the following atomic norm minimization problem in step 2):

[0106] (6)

[0107] In the formula, atomic norm As shown below:

[0108] (7)

[0109] Represents the trace of a matrix, " denotes the Frobenius norm; This is the upper bound of the noise Frobenius norm.

[0110] Example 6:

[0111] The multi-frequency synchronous two-dimensional meshless compressed beamforming method, with the same technical content as any one of embodiments 1-5, further, in step 2), the equivalent dual problem is as follows:

[0112] (8)

[0113] In the formula, for The List, It is a Lagrange multiplier.

[0114] Example 7:

[0115] The multi-frequency synchronous two-dimensional meshless compressed beamforming method, with the same technical content as any one of embodiments 1-6, further includes the following steps in step 2), where the atomic norm minimization problem is transformed into an equivalent dual problem:

[0116] Step 2.1) Construct the Lagrangian function for the atomic norm minimization problem. ,Right now:

[0117] (9)

[0118] in, and For Lagrange multipliers, This indicates the conjugate transpose. Indicates taking the real part;

[0119] Step 2.2) Convert the Lagrange function The maximum lower bound as the dual function ,Right now:

[0120] (10)

[0121] In the formula, "inf" represents the infimum;

[0122] Step 2.3) Solve ,get:

[0123] (11)

[0124] Step 2.4) Construct the optimal noise matrix ;

[0125] exist Calculate the dual function at point, and at Maximizing the top yields:

[0126] (12)

[0127] Step 2.5) Construct the optimal dual variables ;

[0128] Determine the optimal noise matrix and optimal dual variables The dual function at that point, i.e.:

[0129] (13)

[0130] Step 2.6) Determine Lower bound, construct equivalent dual problem.

[0131] because ,in, This represents the modulo operation on each element of the matrix. for 3D identity matrix for and The element-level angle between them. When hour, Non-negative, lower bound is 0; when At that time, the lower boundary is Therefore, the dual problem is:

[0132] (14)

[0133] Example 8:

[0134] The multi-frequency synchronous two-dimensional meshless compressed beamforming method, with the same technical content as any one of embodiments 1-7, further wherein, in step 3), the semidefinite programming problem is as follows:

[0135] (15)

[0136]

[0137] in, Let be a Hermitian matrix, satisfying ; Represents the first of the matrix A column vector consisting of diagonal elements. Indicates the diagonal offset. The main diagonal, The time indicates the first one above the main diagonal. Secondary diagonal, The time indicates the first line below the main diagonal. Secondary diagonal; This indicates a summation operation on all elements of the vector or matrix within the parentheses;

[0138] Example 9:

[0139] The method for multi-frequency synchronous two-dimensional meshless compressed beamforming is the same as any one of embodiments 1-8. Further, in step 3), the semidefinite programming problem is solved by the SDPT3 solver in the CVX toolbox.

[0140] Example 10:

[0141] The multi-frequency synchronous two-dimensional meshless compressed beamforming method has the same technical content as any one of embodiments 1-9. Further, in step (4), the step of estimating the DOA of the sound source and quantizing the intensity includes:

[0142] Step 4.1) Define the spatial spectral function, i.e.:

[0143] (16)

[0144] In the formula, for The List; For spatial spectral functions;

[0145] Step 4.2) Mid-peak value greater than The number of peaks is denoted as ,Will Each peak position is used as Estimated location of each sound source ;

[0146] Step 4.3) Calculate based on the estimated DOA of the sound source. and And construct a perception matrix ;

[0147] Strength calculated using the least squares method ,Right now:

[0148] (17)

[0149] In the formula, It indicates a false reversal.

[0150] Example 11:

[0151] The method for multi-frequency synchronous two-dimensional meshless compressed beamforming includes the following steps:

[0152] Step 1) Establish a two-dimensional multi-frequency synchronization model for the microphone array to acquire signals.

[0153] Continuous expression of multi-frequency sound source signals for:

[0154] (1)

[0155] in, , , express DOA of the sound source and These represent the elevation angle and the azimuth angle, respectively. , They represent in shaft and The distance between two adjacent microphones along the axial direction. For the Dirac function, Indicates each frequency The row vector formed by the sound pressure generated by the sound source at the origin of the coordinate system. Indicates the sound source index. Indicates the total number of sound sources. Indicates frequency index, This represents the total number of frequencies.

[0156] The microphone array is a uniform rectangular array, containing OK Total A microphone, indicated by the symbol "●". Indicates the index of each microphone. The microphone is located at the origin. Let the row vector... For sound sources at various frequencies The sound pressure at the microphone. It can be written as:

[0157] (2)

[0158] in, ,symbol" " represents the Hadamard product, For the speed of sound, It is a row vector composed of the center frequencies of discrete multi-frequency sub-bands. , This indicates the formation of a diagonal matrix with the vector inside the brackets as its diagonal.

[0159] Use the symbol " " represents the Khatri-Rao product, constructing a matrix :

[0160] (3)

[0161] Brief Notes for , ,in for The Column. Construct a matrix. , Indicates transpose. It has the following expression:

[0162] (4)

[0163] in, , Noise interference exists. At that time, the sound pressure was measured. It can be represented as:

[0164] (5)

[0165] The signal-to-noise ratio is defined as ,in" " represents the Frobenius norm.

[0166] Step 2: Establish the atomic norm minimization problem and its equivalent dual problem.

[0167] To estimate the DOA in continuous angular space, we establish the following atomic norm minimization problem:

[0168] (6)

[0169] in express atomic norm,

[0170] (7)

[0171] Represents the trace of a matrix. This is the upper bound of the noise Frobenius norm.

[0172] because Since the parameter is continuous, the original problem in (6) is an infinite-dimensional optimization problem that cannot be solved directly. Therefore, it is transformed into an equivalent dual problem, namely a finite-dimensional optimization problem. First, the Lagrangian function of equation (7) is constructed:

[0173] (8)

[0174] in, and For Lagrange multipliers, This indicates the conjugate transpose. This indicates taking the real part. Dual function. The maximum lower bound of equation (8) is:

[0175] (9)

[0176] Here, "inf" represents the infimum.

[0177] In the noise matrix Minimize dual function That is, by solving get:

[0178] (10)

[0179] Obtain the optimal noise matrix .exist Calculate the dual function at point, and at Maximize:

[0180] (11)

[0181] Obtaining the optimal dual variable At the optimal value and The dual function at point is:

[0182] (12)

[0183] because ,in, This represents the modulo operation on each element of the matrix. for 3D identity matrix for and The element-level angle between them. Therefore, when hour, Non-negative, lower bound is 0; when At that time, the lower boundary is Therefore, the dual problem is:

[0184] (13)

[0185] Step 3: Construct a semidefinite programming problem and solve the dual problem.

[0186] The dual problem shown in equation (12) can be rewritten as the following positive semidefinite programming problem.

[0187] (14)

[0188]

[0189] in, Let be a Hermitian matrix, satisfying . This indicates a summation operation on all elements of the vector or matrix within the parentheses. Represents the first of the matrix A column vector consisting of diagonal elements. Indicates the diagonal offset. The time is the main diagonal. The time indicates the first one above the main diagonal. Secondary diagonal, The time indicates the first line below the main diagonal. Diagonal of the strip.

[0190] The semidefinite programming problem shown in equation (14) can be solved efficiently using the SDPT3 solver in the CVX toolbox.

[0191] Step 4: Estimate the DOA of the sound source and quantize the intensity.

[0192] Define the spatial spectrum function

[0193] (15)

[0194] in for The Based on the relationship between the primal problem and the dual problem, The peak position corresponds to the sound source position. Mid-peak value greater than The number of peaks is denoted as Corresponding Each peak position represents Estimated location of each sound source .

[0195] Based on the estimated DOA of the sound source, we obtain and Then construct the perception matrix. Then calculate using the least squares method. :

[0196] (16)

[0197] in It indicates a false reversal.

[0198] The simulation experiment is as follows:

[0199] To verify the accuracy of this invention, a sound source identification simulation was performed. The specific process is as follows:

[0200] 1. Assuming the location and intensity of the sound source, simulate the sound pressure received by the microphone array in a forward direction;

[0201] 2. Based on step 2, construct the atomic norm minimization problem and the dual problem;

[0202] 3. Based on step 3, transform the problem into a semidefinite programming problem and solve the dual problem;

[0203] 4. Identify the sound source according to step 4;

[0204] The simulation settings are as follows: The uniform rectangular array used in the simulation has 8 rows and 8 columns, for a total of 64 microphones, with the microphone spacing as follows. Its applicable upper frequency limit is approximately 4900 Hz. Assuming three sound sources, the radiated sound waves contain not only a 7000 Hz frequency component, but also 7500 Hz and 8000 Hz frequency components. The intensities of each frequency from the three sound sources are [100 94 94] dB, [94 100 94] dB, and [97 97 94] dB, respectively (reference). ), DOA ( ) are respectively , and Gaussian white noise with an SNR of 20 dB is added when measuring sound pressure using a forward analog microphone array.

[0205] Simulation results: At an aliasing frequency of 7000 Hz, all three sound sources were accurately located. The method proposed in this invention can effectively suppress spatial aliasing and accurately identify sound sources with wavelengths less than twice the minimum microphone spacing.

Claims

1. A multi-frequency synchronous two-dimensional meshless compressed beamforming method, characterized in that, Includes the following steps: Step 1) Establish a two-dimensional multi-frequency synchronization model for the microphone array to acquire signals; Step 2) Based on the two-dimensional multi-frequency synchronization model, establish the atomic norm minimization problem and transform it into an equivalent dual problem; Step 3) Transform the equivalent dual problem into a semidefinite programming problem and solve it. Construct a spatial spectral function based on the solution of the semidefinite programming problem. ; Step 4) Based on Estimate the DOA of the sound source and quantize its intensity.

2. The multi-frequency synchronous two-dimensional meshless compressed beamforming method according to claim 1, characterized in that, The microphone array is a uniform rectangular array.

3. The multi-frequency synchronous two-dimensional meshless compressed beamforming method according to claim 1, characterized in that, Step 1), the steps for establishing a two-dimensional multi-frequency synchronization model for the microphone array to acquire signals include: Step 1.1) Construct a continuous representation of the multi-frequency sound source signal ,Right now: (1) in, , , express DOA of the sound source and These represent the elevation angle and the azimuth angle, respectively. , They represent in shaft and The distance between two adjacent microphones along the axial direction. For the Dirac function, Indicates each frequency The row vector formed by the sound pressure generated by the sound source at the origin of the coordinate system. Indicates the sound source index. Indicates the total number of sound sources. Indicates frequency index, Indicates the total number of frequencies; Step 1.2) Construct sound sources at various frequencies in The sound pressure vector at the microphone ,Right now: (2) in, Indicates the index of each microphone. ,symbol" " represents the Hadamard product, For the speed of sound, It is a row vector composed of the center frequencies of discrete multi-frequency sub-bands. , This indicates the formation of a diagonal matrix with the vectors inside the brackets as its diagonal lines; Step 1.3) Construct the matrix ,Right now: (3) In the formula, the symbol " " represents the Khatri-Rao product; Step 1.4) Let Construct the sound pressure expression, that is: (4) in, , Sound pressure ; Step 1.5) Introduce noise interference into the sound pressure expression to construct a sound pressure measurement module. The expression is used as a two-dimensional multi-frequency synchronization model for the signals acquired by the microphone array; Measuring sound pressure The expression is as follows: (5) In the formula, This is due to noise interference.

4. The multi-frequency synchronous two-dimensional meshless compressed beamforming method according to claim 3, characterized in that, The origin of the coordinate system is The location of the microphone.

5. The multi-frequency synchronous two-dimensional meshless compressed beamforming method according to claim 1, characterized in that, In step 2), the atomic norm minimization problem is as follows: (6) " " denotes the Frobenius norm; This is the upper bound of the noise Frobenius norm; in, atomic norm As shown below: (7) In the formula, Represents the trace of a matrix.

6. The multi-frequency synchronous two-dimensional meshless compressed beamforming method according to claim 1, characterized in that, In step 2), the equivalent dual problem is as follows: (8) In the formula, for The List, It is a Lagrange multiplier.

7. The multi-frequency synchronous two-dimensional meshless compressed beamforming method according to claim 1, characterized in that, Step 2) involves transforming the atomic norm minimization problem into an equivalent dual problem, including: Step 2.1) Construct the Lagrangian function for the atomic norm minimization problem. ,Right now: (9) in, and For Lagrange multipliers, This indicates the conjugate transpose. Indicates taking the real part; Step 2.2) Convert the Lagrange function The maximum lower bound as the dual function ,Right now: (10) In the formula, "inf" represents the infimum; Step 2.3) Solve ,get: (11) Step 2.4) Construct the optimal noise matrix ; exist Calculate the dual function at point, and at Maximizing the top yields: (12) Step 2.5) Construct the optimal dual variables ; Determine the optimal noise matrix and optimal dual variables The dual function at that point, i.e.: (13) Step 2.6) Determine The lower bound is used to construct the equivalent dual problem. The steps include: Step 2.6.1) Construct the function lower bound expression ;in, This represents the modulo operation on each element of the matrix. for 3D identity matrix for and The element-level angle between them; when hour, Non-negative, lower bound is 0; when At that time, the lower boundary is ; Step 2.6.2) Based on function The lower bound expression determines the dual problem, i.e.: (14) In the formula, Represents the trace of a matrix.

8. The multi-frequency synchronous two-dimensional meshless compressed beamforming method according to claim 1, characterized in that, In step 3), the semidefinite programming problem is as follows: (15) in, Let be a Hermitian matrix, satisfying ; Represents the first of the matrix A column vector consisting of diagonal elements. Indicates the diagonal offset. The main diagonal, The time indicates the first one above the main diagonal. Secondary diagonal, The time indicates the first line below the main diagonal. Secondary diagonal; This indicates that the summation operation is performed on all elements of the vector or matrix within the parentheses.

9. The multi-frequency synchronous two-dimensional meshless compressed beamforming method according to claim 1, characterized in that, In step 3), the semidefinite programming problem is solved using the SDPT3 solver in the CVX toolbox.

10. The multi-frequency synchronous two-dimensional meshless compressed beamforming method according to claim 1, characterized in that, Step 4), which involves estimating the DOA of the sound source and quantizing its intensity, includes: Step 4.1) Define the spatial spectral function, i.e.: (16) In the formula, for The List; For spatial spectral functions; Step 4.2) Mid-peak value greater than The number of peaks is denoted as ,Will Each peak position is used as Estimated location of each sound source ; Step 4.3) Calculate based on the estimated DOA of the sound source. and And construct a perception matrix ; Strength calculated using the least squares method ,Right now: (17) In the formula, It indicates a false reversal.

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