阅读说明：本技术 基于原子范数的球面阵列声源波达方向估计方法 (Spherical array sound source direction of arrival estimation method based on atomic norm ) 是由 褚志刚 杨洋 于 2020-06-16 设计创作，主要内容包括：本发明公开了基于原子范数的球面阵列声源波达方向估计方法,步骤为：1)搭建由Q个传声器(2)构成的球形传声器(2)阵列；2)声源(1)向球形传声器(2)阵列辐射声波；3)建立声源波达方向测量模型,并构建传声器(2)测量得到的声压信号矩阵P<Sup>★</Sup>；4)建立协方差矩阵<Image he="80" wi="260" file="DDA0002540654490000011.GIF" imgContent="drawing" imgFormat="GIF" orientation="portrait" inline="no"></Image>5)利用球面ESPRIT算法对协方差矩阵<Image he="81" wi="230" file="DDA0002540654490000012.GIF" imgContent="drawing" imgFormat="GIF" orientation="portrait" inline="no"></Image>进行解算,确定声源波达方向。本发明能克服球面ESPRIT在高频、相干声源或少数据快拍工况下失效的缺陷,并显著提高低SNR工况下的声源DOA估计精度,即使在存在混响的普通测试环境中,本发明仍然有效。(The invention discloses a spherical array sound source direction of arrival estimation method based on atomic norm, which comprises the following steps: 1) building a spherical microphone (2) array consisting of Q microphones (2); 2) a sound source (1) radiates sound waves to a spherical microphone (2) array; 3) establishing a sound source direction of arrival measurement model and constructing a sound pressure signal matrix P obtained by measuring a microphone (2) ★ (ii) a 4) Establishing a covariance matrix 5) Covariance matrix pair using spherical ESPRIT algorithm And resolving is carried out, and the arrival direction of the sound source is determined. The method can overcome the defect that the spherical ESPRIT fails under the working conditions of high frequency, coherent sound source or few data snapshots, obviously improve the DOA estimation precision of the sound source under the working condition of low SNR, and is still effective even in the common test environment with reverberation.)

1. The method for estimating the direction of arrival of the sound source of the spherical array based on the atomic norm is characterized by comprising the following steps of:

1) building a spherical microphone (2) array formed by the Q microphones (2); the center of the array of the spherical microphone (2) is marked as a coordinate origin; wherein the position of the q-th microphone (2) is recorded as (a, omega)_Mq) (ii) a a is the array radius, Q is 1,2, …, Q; the omega is (theta, phi) represents any direction in the three-dimensional space where the spherical microphone (2) array is located; theta is equal to 0 DEG and 180 DEG]Is the elevation angle, phi belongs to [0 DEG, 360 DEG ] is the azimuth angle;

2) the sound source (1) radiates sound waves to the array of spherical microphones (2).

3) Establishing a sound source direction of arrival measurement model and constructing a sound pressure signal matrix P obtained by measuring a microphone (2)^★；

4) Establishing a covariance matrix

5) Covariance matrix pair using spherical ESPRIT algorithmAnd resolving is carried out, and the arrival direction of the sound source is determined.

2. The method for estimating the direction of arrival of a sound source of a spherical array based on an atomic norm as claimed in claim 1 or 2, wherein: the method for establishing the sound source direction of arrival measurement model comprises the following steps:

1) calculating the transfer function t ((ka, omega) from the ith sound source to the qth microphone (2)_Mq)|Ω_Si) Namely:

in the formula, n and m are the order and the order of the spherical harmonic function respectively; b_n(ka) is the modal intensity;

modal intensity b_n(ka) is as follows:

in the formula, j_n(ka) is an n-th order Bessel function of the first kind,is a second class ball Hankel function of order n; j'_n(ka) and

2) calculating spherical harmonic function in omega directionNamely:

in the formula (I), the compound is shown in the specification,is a function of legendre; beta is a_n,m,κIs the associated Legendre function coefficient; spherical harmonic coefficient corresponding to nth order m-order item of spherical harmonic function

3) Establishing a vector of the directions of the Q microphones (2)And corresponding spherical harmonic vector

4) Establishing a transfer function matrix from each sound source to all microphones (2) and recording

wherein, the spherical harmonic function matrix corresponding to the direction of the microphoneAs follows:

in the formula, N₀Infinity represents the highest order of the spherical harmonic;

spherical harmonic function matrix corresponding to direction of sound sourceAs follows:

array modal intensity matrixAs follows:

5) establishing a sound pressure signal matrix measured by each microphone (2)

in the formula (I), the compound is shown in the specification,

6) to spherical harmonic function matrixSpherical harmonic function matrixAnd modal intensity matrix

updating the sound pressure signal matrix P based on the highest order N^★The following were used:

3. the method for estimating the direction of arrival of a sound source in a spherical array based on atomic norms as claimed in claim 1 or 2, wherein a covariance matrix is established

1) establishing an atomic norm minimization model, comprising the following steps:

1.1) establishing a joint Legendre function expression, namely:

wherein x is a function input;

wherein, the function term (x) is connected with Legendre²-1)ⁿAs follows:

in the formula, polynomial coefficient

1.2) substituting the formula (11) into the formula (9) to obtain a legendre function expression when the legendre function power m is more than or equal to 0, namely:

1.3) determining the sine function sin θ ═ (e) for the elevation angle θ^jθ-e^-jθ) /(2j) and cosine function cos θ ═ e^jθ+e^-jθ)/2；

1.4) based on step 1.2) and step 1.3), the associated legendre function expression is updated as follows:

1.5) associated Legendre function term (e)^jθ-e^-jθ)^mAssociated Legendre function term (e)^jθ+e^-jθ-2)^o-mRespectively as follows:

1.6) based on the formula (14) and the formula (15), updating the associated Legendre function expression as follows:

1.7) given the function order n and the number m of stages, let the index o increase from m to n, the index u increase from 0 to m, the index v increase from 0 to o-m, the index w increase from 0 to o-m-v;

determining, for each group (o, u, v, w), an index k of 2u + v-m-w in the trigonometric polynomial expansion associated with the legendre function and coefficients in the trigonometric polynomial of the associated legendre function according to equation (16); after all groups (o, u, v, w) are calculated, the coefficients corresponding to the same kappa value are added to obtain beta_n,m,κ；

1.8) combining equation (9) with Legendre function coefficient beta when m is greater than or equal to 0_n,m,κDetermining the associated Legendre function coefficient beta when m < 0_n,m,κ；

1.9) determining the associated Legendre function coefficient beta_n,m,κThen, spherical harmonic function in omega direction is constructed

1.10) constructing the matrixSum matrix

In the formula, the elements in the matrix D d_θ(θ_Si)、d_φ(φ_Si) Is the basis vector that constitutes matrix D;

memory vectorThe (N + kappa) (2N +1) + N-m +1 element of (A) is_n,mβ_n,m,κAnd k is-n, …,0, …, n, and let the other elements be 0, then the transposed conjugate of the spherical harmonic matrix is obtainedThe sound pressure signal matrix P measured by each microphone (2)^★≈Y_MNB_NGDS+N；

1.11) to establish an input matrix X, namely:

in the formula (I), the compound is shown in the specification,

the atomic norm of equation (18) is as follows:

in the formula, "inf" represents an infimum limit;

collection of atomsAs follows:

1.12) establishing an atomic norm minimization model, namely:

wherein, the noise control parameter is the noise control parameter;

2) establishing an equivalent semi-positive planning model, comprising the following steps:

2.1) converting equation (21) into the following semi-definite programming model:

wherein u and E are auxiliary amounts; n is a radical of_uIs the number of elements in the auxiliary quantity u; t is_b(. h) is the dual Toeplitz operator; for any given vector

2.2) Using the Dual Toeplitz operator T_b(u) mapping u to a block Toeplitz type Hermitian matrix of (2N +1) × (2N +1) dimensions, i.e.:

in the formula, each matrix is partitioned into blocks T_κToeplitz matrices, both of (2N +1) × (2N +1) dimensions: kappa is more than or equal to 0 and less than or equal to 2N;

matrix block T_κAs follows:

23) building a matrix

the van dermonde decomposition formula is shown below:

where the matrix V ═ d (Ω)_S1),d(Ω_S2),…,d(Ω_Sr)](ii) a The diagonal matrix Σ ═ diag ([ σ ])₁,σ₂,…,σ_r])；i is 1,2, …, r; r is a matrix

3) solving the semi-definite planning model by using an alternating direction multiplier method, comprising the following steps of:

3.1) updating a semi-definite programming model by using an alternative direction multiplier method to obtain:

in the formula, Z is an auxiliary matrix, and tau is a regularization parameter;

3.2) establishing an augmented Lagrangian function expression of formula (26), namely:

in the formula (I), the compound is shown in the specification,is a Hermitian Lagrange multiplier matrix; rho > 0 is a penalty parameter; "<·,·>"represents the inner product;

3.3) solving the formula (26) by using an alternative direction multiplier method, and initializing an auxiliary matrix Z⁰＝Λ⁰The variables at γ +1 iterations are updated to be 0:

3.4) dividing the Hermitian Lagrange multiplier matrix and the auxiliary matrix to obtain:

3.5) based on step 3.3) and step 3.4), update equation (28) as follows:

in the formula I₁And I₂Respectively L X L and (2N +1)²×(2N+1)²A dimension unit matrix;is T_bThe companion operator of (·). For any given matrix

3.6) based on step 3.3), update equation (29) as follows:

equation (36) represents the Hermitian matrix

4) based on the sum of the covariance matrix

4. The method for estimating the direction of arrival of a sound source of a spherical array based on atomic norms according to claim 1, wherein determining the direction of arrival of the sound source comprises the following steps:

1) eigenvalue decompositionAnd the eigenvectors are arranged in descending order according to the magnitude of the eigenvalue; writing the first s eigenvectors into the matrix U_SPerforming the following steps;

2) based on the spherical harmonic recursive relation, a matrix U is established_SA related linear equation system, and a least square method is adopted to pair the matrix U_SSolving a related linear equation set to obtain a conversion matrix containing the direction of arrival of the sound source;

the system of linear equations is shown below:

wherein the content of the first and second substances,

wherein the coefficient matrix

3) and performing eigenvalue decomposition on the matrix containing the sound source direction of arrival by adopting a generalized eigenvalue decomposition method to determine the sound source direction of arrival.

Technical Field

The invention relates to the field of sound source identification, in particular to a spherical array sound source direction of arrival estimation method based on atomic norm.

Background

The problem Of Direction-Of-Arrival (DOA) estimation generally exists in the fields Of noise source identification, target detection, fault diagnosis and the like. The rotation invariant Signal parameter Estimation (ESPRIT) technology based on spherical microphone array measurement, referred to as spherical ESPRIT for short, is focused by the advantages of panoramic estimation of the sound source direction of arrival, low computational complexity and the like. The spherical ESPRIT takes a covariance matrix of measurement signals of an array microphone as input, and converts a sound source DOA estimation problem into a least square solution and eigenvalue decomposition problem based on a recursion relation of a spherical harmonic function. The existing spherical ESPRIT is only suitable for the situation that the spherical harmonic function sampled by the microphone satisfies orthogonality, and the discreteness of the microphone makes the order of the spherical harmonic function satisfying the characteristics not high, which limits the available upper limit frequency. Secondly, as a subspace approach, the spherical ESPRIT inevitably suffers from the inherent defect that the subspace approach fails for coherent sound sources, small number of data snapshots or low Signal-to-Noise Ratio (SNR) conditions. These problems and deficiencies constitute key obstacles that prevent the success of spherical ESPRIT in solving the various acoustic source DOA estimation problems. For the defect of limited upper limit frequency, no relevant report for solving the method is found at present.

Aiming at the defects of coherent sound source, snapshot of a few data or failure of low SNR working condition, the prior art adopts a forward and reverse average method and a frequency smoothing technology to remove the source correlation. However, the forward and backward averaging method is only suitable for the case where the number of coherent sound sources is 2, and the frequency smoothing technique is only suitable for the case where the sound sources radiate a broadband signal.

In summary, spherical ESPRIT is still open to the failure of high frequency sound sources, coherent sound sources, snapshot of few data, or low SNR conditions.

Disclosure of Invention

The present invention is directed to solving the problems of the prior art.

The technical scheme adopted for achieving the purpose of the invention is that the method for estimating the direction of arrival of the spherical array sound source based on the atomic norm comprises the following steps:

1) and building a spherical microphone array consisting of Q microphones. The spherical microphone array center is noted as the origin of coordinates. Wherein the qth microphone position is noteda is the array radius, Q is 1,2, …, Q. Omega-theta, phi represents any direction in the three-dimensional space of the spherical microphone array。θ∈[0°,180°]For elevation, φ ∈ [0 °,360 °) is the azimuth.

2) The sound source radiates sound waves towards the spherical microphone array.

3) Establishing a sound source direction of arrival measurement model and constructing a sound pressure signal matrix P obtained by microphone measurement^★。

The method for establishing the sound source direction of arrival measurement model comprises the following steps:

3.1) calculating the transfer function from the ith sound source to the qth microphone

Namely:

in the formula, n and m are the order and the order of the spherical harmonic function, respectively. b_n(ka) is the modal intensity.

Is a spherical harmonic function in the omega direction. Omega_SiIndicating the direction of arrival of the ith sound source. I is 1,2, …, I. I is the total number of sound sources. k is the wave number of the sound wave radiated by the sound source. Superscript denotes conjugation.

Modal intensity b_n(ka) is as follows:

in the formula, j_n(ka) is an n-th order Bessel function of the first kind,is a second class of spherical hank functions of order n. j'_n(ka) andare respectively n-order Bessel functions j of the first kind_n(ka) and n-th order second class spherical Hank functionThe first derivative of (a). The open sphere means that the microphone is arranged on the surface of the open sphere. A rigid sphere means that the microphone is arranged on the surface of a rigid sphere.

3.2) calculating spherical harmonics in the omega directionNamely:

in the formula (I), the compound is shown in the specification,is a function of legendre. Beta is a_n,m,κAlong with legendre function coefficients. Spherical harmonic coefficient corresponding to nth order m-order item of spherical harmonic function

3.3) establishing the vector of the directions of the Q microphonesAnd corresponding spherical harmonic vectorEstablishing vectors of directions of I sound sources

And corresponding spherical harmonic vector

3.4) establishing a transfer function matrix of each sound source to all microphones, notedThe superscript H denotes the transposed conjugate.

Wherein, the spherical harmonic moment corresponding to the direction of the microphoneMatrix of

As follows:

in the formula, N₀Infinity represents the highest order of the spherical harmonics.

Spherical harmonic function matrix corresponding to direction of sound sourceAs follows:

array modal intensity matrix

As follows:

3.5) establishing a sound pressure signal matrix obtained by measuring each microphoneNamely:

in the formula (I), the compound is shown in the specification,is a noise matrix. Signal-to-noise ratio SNR is 20lg (| | P)^★-N||_F/||N||_F). Sound source intensity matrixAnd L is the total number of snapshots. | | non-woven hair_FRepresents the Frobenius norm. Upper label^★The measured quantity is indicated.

3.6) matrix of spherical harmonics

Spherical harmonic function matrixAnd modal intensity matrix

By truncation of order n, i.e. by making a spherical harmonic matrix

Spherical harmonic function matrixAnd modal intensity matrixHighest order of (2)

Indicating an integer that rounds the value to the second nearest toward positive infinity.

Updating the sound pressure signal matrix P based on the highest order N^★The following were used:

4) establishing a covariance matrix

Establishing a covariance matrixThe method comprises the following steps:

4.1) establishing an atomic norm minimization model, comprising the following steps:

4.1.1) establishing a joint Legendre function expression, namely:

in the formula, x is a function input.

Wherein, the function term (x) is connected with Legendre²-1)ⁿAs follows:

in the formula, polynomial coefficient

4.1.2) substituting the formula (11) into the formula (9) to obtain a conjunctive Legendre function expression when the conjunctive Legendre function power m is more than or equal to 0, namely:

4.1.3) determining the sine function sin θ ═ (e) for the elevation angle θ^jθ-e^-jθ) /(2j) and cosine function cos θ ═ e^jθ+e^-jθ)/2。

4.1.4) based on step 4.1.2) and step 4.1.3), the associated legendre function expression is updated as follows:

4.1.5) associated Legendre function term (e)^jθ-e^-jθ)^mAssociated Legendre function term (e)^jθ+e^-jθ-2)^o-mRespectively as follows:

4.1.6) based on equation (14) and equation (15), the associated legendre function expression is updated as follows:

4.1.7) given a function order n and a number of stages m, let index o increase from m to n, index u increase from 0 to m, index v increase from 0 to o-m, index w increase from 0 to o-m-v.

For each set (o, u, v, w), the index k in the trigonometric polynomial expansion with legendre function is determined as 2u + v-m-w and the coefficients in the trigonometric polynomial with legendre function according to equation (16). After all groups (o, u, v, w) are calculated, the coefficients corresponding to the same kappa value are added to obtain beta_n,m,κ。

4.1.8) associated with formula (9) and with Legendre function coefficient beta when m is greater than or equal to 0_n,m,κDetermining the associated Legendre function coefficient beta when m < 0_n,m,κ。

4.1.9) determining the associated Legendre function coefficient beta_n,m,κThen, spherical harmonic function in omega direction is constructedNamely:

4.1.10) construction of matricesSum matrix

In the formula, the elements in the matrix D

The (N + kappa) (2N +1) + N-m +1 elements are marked as A_n,mβ_n,m,κAnd k is-n, …,0, …, n, and let the other elements be 0, then the transposed conjugate of the spherical harmonic matrix is obtained

Sound pressure signal matrix P measured by each microphone^★≈Y_MNB_NGDS+N。

4.1.11) establishes an input matrix X, i.e.:

in the formula (I), the compound is shown in the specification,

is row i of S.

||ψ_i||₂＝1。

The atomic norm of equation (18) is as follows:

in the formula, "inf" represents an infimum limit.Is a collection of atoms.

Collection of atomsAs follows:

4.1.12) to establish an atomic norm minimization model, namely:

among these are noise control parameters.

Is a measure of the sparsity of sound sources in the continuous domain.

Representing the optimal solution to the atomic norm minimization problem.

4.2) establishing an equivalent semi-positive definite planning model, comprising the following steps:

4.2.1) converting equation (21) into the following semi-positive planning model:

wherein u and E are auxiliary amounts. N is a radical of_uIs the number of elements in the auxiliary quantity u. T is_b(. cndot.) is the dual Toeplitz operator. For any given vector Is a half space of (2N,2N), N_u＝8N²+4N+1。

4.2.2) Using the Dual Toeplitz operator T_b(u) mapping u to a block Toeplitz type Hermitian matrix of (2N +1) × (2N +1) dimensions, i.e.:

in the formula, each matrix is partitioned into blocks T_κToeplitz matrices, both of (2N +1) × (2N +1) dimensions: kappa is more than or equal to 0 and less than or equal to 2N.

Matrix block T_κAs follows:

4.2.3) building matricesVandermonde decomposition of (a), making equation (21) and equation (22) equivalent;

the van dermonde decomposition formula is shown below:

where the matrix V ═ d (Ω)_S1),d(Ω_S2),…,d(Ω_Sr)](ii) a The diagonal matrix Σ ═ diag ([ σ ])₁,σ₂,…,σ_r])；

i is 1,2, …, r; r is a matrix

The rank of (d); r.ltoreq.2N +1 is a sufficient condition for the existence of formula (25). Matrix array

The sum of the covariance matrices of the signals due to the individual sources in a group of sources is considered, excluding the covariance components between the signals due to different sources.

4.3) solving the semi-definite planning model by using an alternative direction multiplier method, comprising the following steps:

4.3.1) updating the semi-positive definite programming model by using an alternative direction multiplier method to obtain:

wherein Z is an auxiliary matrix and tau is a regularization parameter.

4.3.2) establish an augmented Lagrangian function expression of equation (26), namely:

in the formula (I), the compound is shown in the specification,

is the Hermitian lagrange multiplier matrix. ρ > 0 is a penalty parameter. "<·,·>"denotes the inner product.

4.3.3) solving the equation (26) by means of the alternative direction multiplier method, initializing the auxiliary matrix Z⁰＝Λ⁰The variables at γ +1 iterations are updated to be 0:

4.3.4) dividing the Hermitian Lagrange multiplier matrix and the auxiliary matrix to obtain:

4.3.5) based on step 3.3) and step 3.4), update equation (28) as follows:

in the formula I₁And I₂Respectively L X L and (2N +1)²×(2N+1)²A dimension unit matrix.

Is T_bThe companion operator of (·). For any given matrixM＝diag([(2N+1)×[2N+1,2N,…,1],

M is a diagonal matrix. Matrix array A basic Toeplitz matrix in which the elements of the k (m) th diagonal are all 1 and the other elements are all 0.

4.3.6) based on step 3.3), update equation (29) as follows:

equation (36) represents the Hermitian matrix

Projection to semipositiveOn the cone, the Hermitian matrix is subjected to eigenvalue decomposition, and all negative eigenvalues are made to be.

4.4) based on the sum of the covariance matrices foundEstablishing a covariance matrix

5) Covariance matrix pair using spherical ESPRIT algorithm

Resolving is carried out, and the arrival direction of the sound source is determined, and the method comprises the following steps:

5.1) eigenvalue decomposition

And sorting the feature vectors in descending order according to the size of the feature values. Writing the first s eigenvectors into the matrix U_SIn (1).

5.2) establishing a matrix U based on the spherical harmonic recursive relationship_SA related linear equation system, and a least square method is adopted to pair the matrix U_SAnd solving the related linear equation system to obtain a conversion matrix containing the arrival direction of the sound source. The system of linear equations is as follows:

wherein the content of the first and second substances,

for subordinate matrix U by superscript (U, v)_SIn which a part of the rows is selected to be combined into a matrix, psi_xy+、ψ_xy-And psi_zFor a transformation matrix containing the direction of arrival of the sound source,and

is a matrix of coefficients.

Wherein the coefficient matrix

Sum coefficient matrix

Respectively as follows:

and 5.3) carrying out eigenvalue decomposition on the matrix containing the sound source direction of arrival by adopting a generalized eigenvalue decomposition method to determine the sound source direction of arrival.

The technical effects of the invention are undoubted, the invention establishes a novel spherical ESPRIT technology based on atomic norm, and analyzes the performance based on simulation and verification tests, and the result shows that ANM + spherical ESPRIT can perfectly overcome the defect that the spherical ESPRIT fails under the working conditions of high frequency, coherent sound source or a few data snapshots, and obviously improves the estimation precision of the DOA of the sound source under the working condition of low SNR, and the invention is still effective even in the common test environment with reverberation. Secondly, the ADMM-based solving algorithm derived in the invention is more efficient than an IPM-based SDPT3 solver.

Drawings

FIG. 1 is a schematic view of a spherical microphone array and a sound source;

FIG. 2 is a sound source identification imaging plot of a single Monte Carlo calculation for different frequencies;

FIG. 2(a) is an imaging plot of the acoustic source for spherical ESPRIT at a frequency of 1000 Hz;

FIG. 2(b) is a sound source imaging plot of ANM + spherical ESPRIT at 1000Hz frequency, ANM solved by an IPM based SDPT3 solver;

FIG. 2(c) is a sound source imaging plot of ANM + spherical ESPRIT at 1000Hz frequency, ANM solved by ADMM-based solution algorithm;

FIG. 2(d) is an imaging plot of the acoustic source for spherical ESPRIT at a frequency of 3000 Hz;

FIG. 2(e) is a sound source imaging plot of ANM + spherical ESPRIT at 3000Hz frequency, ANM solved by an IPM based SDPT3 solver;

FIG. 2(f) is a sound source imaging plot of ANM + spherical ESPRIT at 3000Hz frequency, the ANM being solved by an ADMM-based solution algorithm;

FIG. 2(g) is an imaging plot of the acoustic source for spherical ESPRIT at a frequency of 1000 Hz;

FIG. 2(h) is a sound source imaging plot of ANM + spherical ESPRIT at 1000Hz frequency, ANM solved by an IPM based SDPT3 solver;

FIG. 2(i) is a sound source imaging plot of ANM + spherical ESPRIT at 1000Hz frequency, ANM solved by ADMM-based solution algorithm;

FIG. 3 is a graph of variation of time taken to estimate root mean square error for the DOA of the acoustic source and ANM solution with frequency;

FIG. 3(a) is a graph showing the variation of sigma with frequency for 100 Monte Carlo calculations

FIG. 3(b) is a comparison graph of the time consumption of two ANM solution methods;

FIG. 4 is a sound source identification imaging plot for a single Monte Carlo calculation of different sound source coherence;

FIG. 4(a) is an imaging diagram of a sound source of spherical ESPRIT when the sound sources are not coherent;

FIG. 4(b) is an imaging diagram of an ANM + spherical ESPRIT sound source when the sound sources are not coherent, the ANM being solved by an IPM-based SDPT3 solver;

FIG. 4(c) is an imaging diagram of the sound source for ANM + spherical ESPRIT when the sound sources are not coherent, the ANM being solved by an ADMM-based solution algorithm;

FIG. 4(d) is an imaging plot of a spherical ESPRIT sound source with the source partially coherent;

FIG. 4(e) is an acoustic source imaging plot of ANM + spherical ESPRIT with the acoustic source partially coherent, the ANM being solved by an IPM based SDPT3 solver;

FIG. 4(f) is an acoustic source imaging plot of ANM + spherical ESPRIT with the acoustic source partially coherent, the ANM being solved by an ADMM-based solution algorithm;

FIG. 4(g) is an imaging plot of a spherical ESPRIT sound source when the source is fully coherent;

FIG. 4(h) is an acoustic source imaging plot of ANM + spherical ESPRIT when the acoustic source is fully coherent, the ANM being solved by an IPM based SDPT3 solver;

FIG. 4(i) is an acoustic source imaging plot of ANM + spherical ESPRIT when the acoustic source is fully coherent, the ANM being solved by an ADMM-based solution algorithm;

FIG. 5 is a complementary cumulative distribution function of the root mean square error of the DOA estimation of the sound source for different total number of snapshots and different SNR;

fig. 5(a) is a complementary cumulative distribution function of sound source DOA estimation root mean square error under different total snapshots and different SNRs when T is 2 °, spherical ESPRIT, and 100 monte carlo calculations;

fig. 5(b) is a complementary cumulative distribution function of sound source DOA estimated root mean square error for different total snapshots and different SNRs when T is 2 °, ANM + sphere ESPRIT (ANM is solved by an IPM-based SDPT3 solver), and 100 monte carlo calculations;

fig. 5(c) is a complementary cumulative distribution function of sound source DOA estimation root mean square error for different total number of snapshots and different SNRs when T is 2 °, ANM + sphere ESPRIT (ANM is solved by ADMM-based solution algorithm), and 100 monte carlo calculations;

fig. 5(d) is a complementary cumulative distribution function of the sound source DOA estimated root mean square error under different total snapshots and different SNRs when T is 1 °, the spherical ESPRIT, and 100 monte carlo calculations;

fig. 5(e) is a complementary cumulative distribution function of sound source DOA estimated root mean square error for different total snapshots and different SNRs when T is 1 °, ANM + sphere ESPRIT (ANM is solved by an IPM-based SDPT3 solver), and 100 monte carlo calculations;

fig. 5(f) is a complementary cumulative distribution function of sound source DOA estimation root mean square error for different total number of snapshots and different SNRs when T is 1 °, ANM + spherical ESPRIT (ANM is solved by ADMM-based solution algorithm), and 100 monte carlo calculations;

FIG. 6(a) is a test layout in a semi-anechoic chamber;

FIG. 6(b) is a development view of the three-dimensional space in the semi-anechoic chamber;

FIG. 7 is an imaging of a test sound source in a semi-anechoic chamber;

FIG. 7(a) is an imaging diagram of a sound source with a spherical ESPRIT at a frequency of 1500Hz with a total number of snapshots of 30, with the speaker excited by a steady-state white noise signal;

FIG. 7(b) is a sound source imaging plot with a speaker excited by a steady-state white noise signal, 1500Hz frequency, and ANM + spherical ESPRIT with a total number of snapshots of 30, the ANM being solved by an IPM-based SDPT3 solver;

FIG. 7(c) is a sound source imaging plot with a speaker excited by a steady-state white noise signal, 1500Hz frequency, and ANM + spherical ESPRIT with a total number of snapshots of 30, the ANM being solved by an ADMM-based solution algorithm;

FIG. 7(d) is an image of a sound source with a spherical ESPRIT at a frequency of 3000Hz with a total number of snapshots of 30, the speaker being excited by a steady-state white noise signal;

FIG. 7(e) is a sound source imaging plot with a speaker excited by a steady-state white noise signal, 3000Hz frequency, and with ANM + spherical ESPRIT at a total snapshot count of 30, the ANM being solved by an IPM-based SDPT3 solver;

FIG. 7(f) is a sound source imaging plot with a speaker excited by a steady-state white noise signal, 3000Hz frequency, ANM + spherical ESPRIT with a total number of snapshots of 30, the ANM being solved by an ADMM-based solution algorithm;

FIG. 7(g) is an image of a sound source with a speaker excited by a steady-state white noise signal, 1500Hz frequency, using spherical ESPRIT with a total of 5 snapshots;

FIG. 7(h) is a sound source imaging plot with a speaker excited by a steady-state white noise signal, 1500Hz frequency, and ANM + sphere ESPRIT with a total number of snapshots of 5, the ANM being solved by an IPM-based SDPT3 solver;

FIG. 7(i) is a sound source imaging plot with a speaker excited by a steady-state white noise signal, 1500Hz frequency, and ANM + spherical ESPRIT with a total number of snapshots of 5, the ANM being solved by an ADMM-based solution algorithm;

FIG. 7(j) is an image of a sound source with a spherical ESPRIT at 1500Hz frequency with the total number of snapshots 1, with the speaker excited by a steady-state white noise signal;

FIG. 7(k) is a sound source imaging plot of a speaker excited by a steady-state white noise signal at 1500Hz with ANM + sphere ESPRIT at a total snapshot count of 1, the ANM being solved by an IPM-based SDPT3 solver;

FIG. 7(l) is a sound source imaging plot of a speaker excited by a steady-state white noise signal at 1500Hz with ANM + sphere ESPRIT when the total number of snapshots is 1, the ANM being solved by an ADMM-based solution algorithm;

FIG. 7(m) is an image of a sound source with a spherical ESPRIT at a frequency of 1500Hz with speakers excited by the same pure tone signal and a total number of snapshots of 30;

FIG. 7(n) is a sound source imaging plot of a speaker excited by the same pure tone signal, at 1500Hz, with ANM + spherical ESPRIT at a total snapshot count of 30, the ANM being solved by an IPM-based SDPT3 solver;

FIG. 7(o) is a sound source imaging plot of a speaker excited by the same pure tone signal at 1500Hz with a total snapshot of 30 ANM + spherical ESPRIT, the ANM being solved by an ADMM-based solution algorithm;

FIG. 8(a) is a test layout in a general room;

FIG. 8(b) is an expanded view of a three-dimensional space of a general room;

FIG. 9 is an imaging diagram of a test sound source in a general room;

FIG. 9(a) is an imaging diagram of a sound source with a spherical ESPRIT at a frequency of 1000Hz with a total number of snapshots of 30, with the speaker excited by a steady-state white noise signal;

FIG. 9(b) is a sound source imaging plot with a speaker excited by a steady-state white noise signal, at a frequency of 1000Hz, using ANM + spherical ESPRIT with a total number of snapshots of 30, the ANM being solved by an IPM-based SDPT3 solver;

FIG. 9(c) is a sound source imaging plot with a speaker excited by a steady-state white noise signal, at a frequency of 1000Hz, using ANM + spherical ESPRIT with a total number of snapshots of 30, the ANM being solved by an ADMM-based solution algorithm;

FIG. 9(d) is an imaging plot of a source with a spherical ESPRIT at a frequency of 2000Hz with the speaker excited by a steady-state white noise signal and a total number of snapshots of 30;

FIG. 9(e) is an acoustic source imaging plot with a speaker excited by a steady-state white noise signal, at 2000Hz, using an ANM + spherical ESPRIT with a total number of snapshots of 30, the ANM being solved by an IPM-based SDPT3 solver;

FIG. 9(f) is an image of a sound source with a speaker excited by a steady-state white noise signal, at a frequency of 2000Hz, using ANM + spherical ESPRIT with a total number of snapshots of 30, the ANM being solved by an ADMM-based solution algorithm;

FIG. 9(g) is an imaging diagram of a sound source with a spherical ESPRIT at a frequency of 3000Hz with a total number of snapshots of 30, with the speaker excited by a steady-state white noise signal;

FIG. 9(h) is a sound source imaging plot with a speaker excited by a steady-state white noise signal, 3000Hz frequency, and with ANM + spherical ESPRIT with a total number of snapshots of 30, the ANM is solved by an IPM-based SDPT3 solver;

FIG. 9(i) is a sound source imaging plot with a speaker excited by a steady-state white noise signal, 3000Hz frequency, and an ANM + spherical ESPRIT with a total number of snapshots of 30, the ANM being solved by an ADMM-based solution algorithm;

FIG. 9(j) is an image of a sound source with a spherical ESPRIT at a frequency of 4000Hz with a speaker excited by a steady-state white noise signal and a total number of snapshots of 30;

FIG. 9(k) is a sound source imaging plot with a speaker excited by a steady-state white noise signal, at 4000Hz frequency, using ANM + spherical ESPRIT with a total number of snapshots of 30, the ANM being solved by an IPM-based SDPT3 solver;

FIG. 9(l) is a sound source imaging plot of a speaker excited by a steady-state white noise signal, at 4000Hz, with ANM + spherical ESPRIT at a total snapshot count of 30, the ANM being solved by an ADMM-based solution algorithm;

FIG. 9(m) is an imaging diagram of a sound source with a spherical ESPRIT at a frequency of 5000Hz with a total number of snapshots of 30, with the speaker excited by a steady-state white noise signal;

FIG. 9(n) is a sound source imaging plot with a speaker excited by a steady-state white noise signal, at a frequency of 5000Hz, using ANM + spherical ESPRIT with a total number of snapshots of 30, the ANM being solved by an IPM-based SDPT3 solver;

FIG. 9(o) is a sound source imaging plot with a speaker excited by a steady-state white noise signal, at a frequency of 5000Hz, using ANM + spherical ESPRIT with a total number of snapshots of 30, the ANM being solved by an ADMM-based solution algorithm;

in the figure, a sound source 1 and a microphone 2 are shown.

Detailed Description

The present invention is further illustrated by the following examples, but it should not be construed that the scope of the above-described subject matter is limited to the following examples. Various substitutions and alterations can be made without departing from the technical idea of the invention and the scope of the invention is covered by the present invention according to the common technical knowledge and the conventional means in the field.