A transient sound field reconstruction method based on variational Bayesian approximation augmented Kalman filter
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- ANHUI UNIV
- Filing Date
- 2024-07-02
- Publication Date
- 2026-05-26
Smart Images
Figure CN118861598B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of transient sound field reconstruction technology, and particularly relates to a transient sound field reconstruction method based on an augmented Kalman filter with variational Bayesian approximation. Background Technology
[0002] When the sound field to be reconstructed is close to the sound source, near-field acoustic holography (NAH) is a highly effective method for noise source identification, localization, and sound field visualization. However, traditional NAH can effectively reconstruct steady-state signals at a single frequency but cannot reconstruct transient sound fields. Therefore, Hald and Jean proposed applying Fourier transform and Laplace transform to time-domain acoustic holography for transient sound field reconstruction. However, due to the application of Laplace transform and Fourier transform, the reconstruction process of these methods must be performed at every frequency. To improve computational efficiency, real-time near-field acoustic holography (RT-NAH) has been proposed for sound field reconstruction in the wavenumber domain. While regularized RT-NAH can be used to solve underdetermined inverse problems, it suffers from winding errors due to the fast Fourier transform in two-dimensional space.
[0003] Zhang et al. proposed the Time-Domain Plane Wave Superposition (TDPWSM) method, which can reduce the orbital error. However, it still cannot avoid inverting underdetermined problems during the reconstruction process. While the application of regularization can obtain a stable solution for each calculation, the automatically selected regularization parameters are constantly updated over time, which can cause the selection of regularization parameters to become ineffective, resulting in the reconstruction error increasing over time. Summary of the Invention
[0004] To address the aforementioned technical problems, this invention proposes a transient sound field reconstruction method based on an augmented Kalman filter with variational Bayesian approximation, thereby resolving the issues present in the prior art.
[0005] To achieve the above objectives, this invention provides a transient sound field reconstruction method based on an augmented Kalman filter with variational Bayesian approximation, comprising:
[0006] A holographic surface is set in the sound field to be measured, and the time-domain sound pressure signal is measured based on the holographic surface to obtain the measured sound pressure column vector;
[0007] A virtual source surface is set in the sound field to be measured, and the wavenumber domain is discretized into several wavenumber points based on the virtual source surface;
[0008] Based on the measured sound pressure column vector and several wavenumber points, the time-domain sound pressure signal is modeled using the time-domain plane wave superposition method to obtain the sound pressure signal model;
[0009] Based on the sound pressure signal model, an augmented Kalman filter based on variational Bayes approximation is used to jointly recursively estimate the state vector of the sound field and the pressure time wavenumber spectrum on the virtual source surface to obtain the state vector estimate and the wavenumber spectrum estimate.
[0010] Based on the state vector estimation and wavenumber spectrum estimation, the sound pressure signal at any spatial point in the sound field at each time moment is reconstructed.
[0011] Optionally, the process of modeling the time-domain sound pressure signal using the time-domain plane wave superposition method includes: convolution superposition of the time-domain wavenumber spectrum of the virtual source surface and the time-domain propagation kernel; wherein, the expression is:
[0012]
[0013] In the formula, * denotes temporal convolution, r = (x, y) represents each spatial location in the sound field, and K l It is the wavenumber (k) of the l-th sampling point in the wavenumber domain. x ,k y ), l = 1, 2, ..., L, where L refers to the number of wavenumber domain sampling points; z r , z h and z v Let Δz represent the positions of planes R, H, and V on the z-axis, respectively. hv =z h -z v Δz rv =z r -z v ;ψ hv (K l ,r,Δz hv ,t) and ψ rv (K l ,r,Δz rv ,t) is the time-domain sound pressure wavenumber spectrum P(K) on the virtual source surface. l ,z v ,t) and the time-varying sound pressure P(r,z) on the holographic surface H h ,t) and the time-varying sound pressure P(r,z) on the reconstructed surface R r The time-domain propagation kernel associated with ,t).
[0014] Optionally, the expression for the time-domain sound pressure at any point on the holographic surface at any time is:
[0015] p h (t i )=ψ hv (t i )*P v (t i )
[0016] Where, p h (ti ) represents t i Time-space holographic surface sound pressure matrix, P v (t i ) represents t i The time-instance virtual source surface acoustic pressure wavenumber spectrum matrix, ψ hv (t i ) represents t i The temporal propagation kernel matrix of the time-domain holographic surface and the virtual source surface.
[0017] Optionally, the process of jointly recursively estimating the state vector of the sound field and the pressure time wavenumber spectrum on the imaginary source surface using an augmented Kalman filter based on the variational Bayesian approximation includes:
[0018] At the beginning of the time series, the state vector of the sound field and the pressure time wavenumber spectrum on the virtual source surface are initially estimated, and the initial covariance matrix is set.
[0019] The feature system implementation algorithm is used to transform the multi-input multi-output system of the time-domain plane wave superposition method into a state-space form, and the state-space equation and input-output measurement equation are obtained.
[0020] An augmented state vector is constructed based on the first-order smoothness condition contained in the discrete pressure-time wavenumber spectrum on the virtual source surface;
[0021] The updated state-space equation and input-output measurement equation are obtained based on the pressure-time wavenumber spectrum on the virtual source surface.
[0022] The augmented state vector and the updated state space equation are approximated by a posterior distribution using the standard variational Bayesian method to obtain a separable approximate distribution.
[0023] Optionally, the feature system implementation algorithm is used to transform the multi-input multi-output system of the time-domain plane wave superposition method into a state-space form, resulting in the state-space equations and input-output measurement equations. The expressions for the state-space equations and input-output measurement equations are as follows:
[0024] x(t i+1 )=Ax(t i )+BP v (t i )+q(t i )
[0025] p h (t i )=Cx(t i )+DP v (t i )+r(t i )
[0026] In the formula, q(t) i) is the Gaussian process noise matrix. Q(t i ) represents q(t) i The covariance matrix of r(t). i ) represents the measurement noise matrix. R(t i ) represents r(t) i The covariance matrix of x(t). i+1 ) represents the time step t i+1 The state matrix, P v (t i ) represents the virtual source surface acoustic pressure wavenumber spectrum matrix, p h (t i ) represents the holographic surface sound pressure matrix, and A, B, C and D represent the discretized state, input, output and feedthrough matrices, respectively.
[0027] Optionally, the matrix equation corresponding to the first-order smoothness condition contained in the discrete pressure time wavenumber spectrum on the virtual source surface is:
[0028] P v (t i+1 ) = P v (t i )+q(t i )
[0029] In the formula, P v (t i ) represents the virtual source surface sound pressure wavenumber spectrum.
[0030] Optionally, the posterior distribution approximation of the augmented state vector and the updated state-space equation using the standard variational Bayesian method is as follows:
[0031] p[X a (t i+1 ),R(t i+1 )|p h (t1:t i )]≈f X (X a (t i+1 ))f R (R(t i+1 ))
[0032] In the formula, f X (X a (t i+1 )) and f R (R(t i+1 )) are respectively X a (t i+1 ) and R(t i+1 Probability density function.
[0033] Compared with the prior art, the present invention has the following advantages and technical effects:
[0034] This invention uses the time-domain plane wave superposition method as the reconstruction algorithm and introduces the spatial state equation to apply an augmented Kalman filter for forward iteration, which has the advantages of good computational stability, high computational accuracy and simple implementation.
[0035] Based on the augmented Kalman filter, this invention proposes the variational Bayes approximation, which performs approximate posterior inference with low computational cost and assumes that the posterior form is simpler and easier to handle analytically.
[0036] This invention approximates the joint posterior distribution of spatial state and noise variance by decomposing a free-form distribution. This approximation is formed separately at each time step, resulting in a recursive algorithm where sufficient statistics for spatial state and noise variance are estimated at each step through fixed-point iterations of an augmented Kalman filter.
[0037] The method of this invention can be used for the analysis of steady-state sound fields, unsteady-state sound fields, and transient sound fields. Attached Figure Description
[0038] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. In the drawings:
[0039] Figure 1 This is a schematic diagram showing the positions of the sound source surface, virtual source surface, holographic surface, and reconstruction surface in an embodiment of the present invention;
[0040] Figure 2 The following are specific examples of embodiments of the present invention, wherein Figure (a) shows the theoretical sound pressure of the reconstructed surface at t = 0.98 ms, Figure (b) shows a comparison of the planar distribution of the reconstructed sound pressure at t = 0.98 ms, Figure (c) shows the sound pressure error distribution at t = 0.98 ms, Figure (d) shows the theoretical sound pressure of the reconstructed surface at t = 2.54 ms, Figure (e) shows a comparison of the planar distribution of the reconstructed sound pressure at t = 2.54 ms, and Figure (f) shows the sound pressure error distribution at t = 2.54 ms.
[0041] Figure 3 This invention provides a comparison of the theoretical and reconstructed temporal sound pressure levels of points R1, R2, and R3 on the reconstructed surface of a planar sound source in an embodiment of the invention.
[0042] Figure 4 This is a graph of two time-domain evaluation factors in an embodiment of the present invention. Detailed Implementation
[0043] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.
[0044] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.
[0045] Example 1
[0046] To replace the inversion process of TDPWSM and improve the accuracy of transient sound field reconstruction, this invention proposes an augmented Kalman filter based on variational Bayesian approximation. Using time-domain sound pressure on the holographic surface as input, and employing the time-domain plane wave superposition method for modeling, the augmented Kalman filter based on variational Bayesian approximation is used to jointly and recursively estimate the state vector in the linear state-space model. Furthermore, the pressure-time wavenumber spectrum on the virtual source plane used to calculate the time-domain signal on the reconstruction surface is employed. This method achieves high-precision reconstruction of the transient sound field by obtaining the time-domain sound pressure at any point on the reconstruction surface.
[0047] like Figure 1 As shown, this embodiment provides a transient sound field reconstruction method based on an augmented Kalman filter with variational Bayesian approximation, including the following steps:
[0048] Step 1: Set up a holographic surface H in the sound field to be measured; arrange M microphone arrays evenly distributed on the holographic surface to measure the time-domain sound pressure signal and obtain the measured sound pressure column vector; the sound field to be measured is a transient sound field;
[0049] Step 2: Set up a virtual source surface V in the space of the sound field to be measured, and discretize the wavenumber domain into L wavenumber points;
[0050] Step 3: Model the spatial sound pressure signal using the time-domain plane wave superposition method. The time-domain sound pressure at any point on the holographic surface at any time t can be expressed as the convolution superposition of the time-domain wavenumber spectrum of the virtual source surface and the time-domain propagation kernel:
[0051]
[0052] Where * denotes temporal convolution, r = (x, y) represents each spatial location in the sound field, and K l It is the wavenumber (k) of the l-th sampling point in the wavenumber domain. x ,k y ), l = 1, 2, ..., L, where L refers to the number of wavenumber domain sampling points; z r , z h and z vLet Δz represent the positions of planes R, H, and V on the z-axis, respectively. hv =z h -z v Δz rv =z r -z v ;ψ hv (K l ,r,Δz hv ,t) and ψ rv (K l ,r,Δz rv ,t) is the time-domain sound pressure wavenumber spectrum P(K) on the virtual source surface. l ,z v ,t) and the time-varying sound pressure P(r,z) on the holographic surface H h ,t) and the time-varying sound pressure P(r,z) on the reconstructed surface R r The time-domain propagation kernel associated with t is calculated using the following formulas:
[0053]
[0054] During the measurement process, it is necessary to consider the discrete time points t. i = (i-1)Δt, i = 1, 2, ..., I, where I represents the total number of time steps. The sound pressure level varying with time on each plane is sampled. In the discrete computation, Δt is one time step, and I is the total number of time steps, where i is the index of the discrete time. From the above formula, it can be seen that when t < Δz... hv When / c, the impulse response function G pp (K l ,Δz hv When ,t) is 0, its corresponding time-domain propagation kernel ψ hv (K l ,r,Δz hv ,t i The value is also 0, at which point the sound wave from the virtual source surface V has not yet radiated to the holographic surface H. Let the first time step of the radiated sound wave propagating to the holographic surface be t. i , then t i The sound pressure level of the holographic surface at a given time can be expressed as:
[0055]
[0056] For ease of expression, Δt is omitted in the following formulas.
[0057] At any time t on the holographic surface i The time-domain sound pressure at any point can be represented in matrix form:
[0058] p h (t i )=ψ hv (ti )*P v (t i )
[0059] in,
[0060] p h (t i )=[P(r1,z h ,t i P(r2,z) h ,t i ) … P(r M ,z h ,t i )] T
[0061] P v (t i )=[P(K1,z v ,t i P(K2,z) v ,t i ) … P(K L ,z v ,t i )] T
[0062]
[0063] Step 4: Starting from the initial test time, use an augmented Kalman filter based on variational Bayesian approximation to jointly recursively estimate the state vector and the pressure time wavenumber spectrum on the virtual source plane in the linear state-space model, and form a separable variational approximation for the joint posterior distribution of the state vector and the pressure time wavenumber spectrum on the virtual source plane at each time step. Specifically, the process of jointly recursively estimating the state vector of the sound field and the pressure time wavenumber spectrum on the virtual source surface using an augmented Kalman filter based on variational Bayesian approximation includes: at the beginning of the time series, initial estimations are made of the state vector of the sound field and the pressure time wavenumber spectrum on the virtual source surface, and an initial covariance matrix is set; the multi-input multi-output system of the time-domain plane wave superposition method is transformed into a state-space form using a feature system implementation algorithm, resulting in the state-space equation and input / output measurement equations; an augmented state vector is constructed based on the first-order smoothness condition contained in the discrete pressure time wavenumber spectrum on the virtual source surface; the updated state-space equation and input / output measurement equations are obtained based on the pressure time wavenumber spectrum on the virtual source surface; and a separable approximate distribution is obtained by posterior distribution approximation of the augmented state vector and the updated state-space equation using the standard variational Bayesian method.
[0064] The core of the Kalman filter is its state-space representation framework, which contains two important equations: the state equation and the measurement equation. The former describes the change in the system state at a specific time step relative to the input and the system state at the previous time step, while the latter determines the relationship between the measurement data and the input and system state. The formula can be expressed as:
[0065] x(t i+1 )=Ax(t i )+BP v (t i )+q(t i )
[0066] p h (t i )=Cx(t i )+DP v (t i )+r(t i )
[0067] Where q(t) i ) is the Gaussian process noise matrix. Q(t i ) represents its covariance matrix. r(t) i ) represents the measurement noise matrix. R(t i Let x(t) represent its covariance matrix. i+1 ) represents the time step t i+1 The state matrix, P v (t i ) represents the virtual source surface acoustic pressure wavenumber spectrum matrix, i.e., the input vector matrix in the Kalman filter, p h (t i The holographic surface sound pressure matrix (SPL) represents the output vector in the Kalman filter. A, B, C, and D represent the discretized state, input, output, and feedthrough matrices, respectively. These are composed of Mb×Lb dimension Hankel matrices, where b is typically 1 / 16, and M and L represent the number of holographic measurement points and wavenumber domain sampling points, respectively. The Hankel matrix calculation formula is as follows:
[0068]
[0069] The singular value decomposition formula for the Hankel matrix at point t1 is as follows:
[0070] Φ(0)=USV T
[0071] Further, the calculation formulas for A, B, C, and D are obtained:
[0072] A = S -1 / 2 U T Φ(t1)VS-1 / 2
[0073] B = S -1 / 2 V T W L
[0074] C = W M T US 1 / 2
[0075] D = ψ hv (t1)
[0076] Among them W L Composed of an L×L dimensional identity matrix, W M It consists of an M×M dimensional zero matrix.
[0077] By introducing the Feature Realization Algorithm (ERA) into a multi-input multi-output system using the time-domain plane wave superposition method, the state-space equation and input / output measurement equations can be obtained:
[0078] x(t i+1 )=Ax(t i )+BP v (t i )+q(t i )
[0079] p h (t i )=Cx(t i )+DP v (t i )+r(t i )
[0080] For the state matrix x(t) i ) and virtual source surface sound pressure wavenumber spectrum P v (t i The joint estimation is the augmented matrix:
[0081]
[0082] Among them, X a (t i It follows a Gaussian prior distribution, i.e. F(t i P(t) represents the mean matrix. i ) represents the covariance matrix.
[0083] Assuming the discrete pressure time wavenumber spectrum on the virtual source surface contains a first-order smoothness condition, the corresponding matrix equation is:
[0084] P v (t i+1 ) = P v (ti )+q(t i )
[0085] The state-space equation can then be further expressed as:
[0086] X a (t i+1 ) = A a X a (t i )+q(t i )
[0087] The input-output measurement equation can be further expressed as:
[0088] p h (t i ) = C a X a (t i )+r(t i )
[0089] in
[0090]
[0091] C a =[CD]
[0092]
[0093] The proposed method considers the observation noise variance parameter as... Let be random variables with independent dynamic models. The diagonal covariance matrix R(t) i It consists of these variance parameters, i.e. The independence assumption between the dynamic models of state and variance parameters can be expressed mathematically as follows:
[0094] [X a (t i+1 ),R(t i+1 )|X a (t i ),R(t i )]=[X a (t i+1 )|X a (t i )][R(t i+1 )|R(t i )]
[0095] Calculating the posterior distribution to achieve optimal Bayesian filtering is a recursive problem, difficult to solve using general methods due to its complex integration. To address this recursive estimation problem, we employ a variational Bayesian approximation method based on posterior updates. This approximation method is combined with a heuristic dynamic model of the measured noise variance. The variational Bayesian method allows for simplified computation of the posterior distribution, enabling efficient updates of the augmentation state and noise variance estimates at each time step.
[0096] The joint state matrix X can be obtained using the standard variational Bayesian method. a (t i+1 The estimated distribution of )
[0097] p[X a (t i+1 ),R(t i+1 )|p h (t1:t i )]≈f X (X a (t i+1 ))f R (R(t i+1 ))
[0098] Where f X (X a (t i+1 )) and f R (R(t i+1 )) are respectively X a (t i+1 ) and R(t i+1 The probability density function can be estimated by its expected value as follows:
[0099]
[0100] in
[0101]
[0102] F i+1 =F i+1 +P i+1 C a T (C a P i+1 C a T +R(t i )) -1 (p h (t i )-C a F i+1 )
[0103] P i+1 =Pi+1 -P i+1 C a T (C a P i+1 C a T +R(t i )) -1 C a P i+1
[0104]
[0105] Unlike traditional Kalman filters, this method measures noise variance. The assumption is stochastic, with an independent dynamic model. Its dynamic model is as follows: the prediction step is always for each... parameters and This generates an inverse gamma prediction distribution. The optimal estimate of the state vector is F. i+1 =X a (t i ).
[0106] Step 5, by X a (t i The virtual source surface sound pressure signal P can be obtained at each time sampling point. v (t i Based on the obtained virtual source surface sound pressure at each time point and the transmission relationship between the virtual source surface and the reconstructed surface, the time-domain sound pressure signal is reconstructed.
[0107] As a specific implementation method of this embodiment, the experimental model diagram is as follows: Figure 1 The sound source is a planar plate, with the center of the plate as the origin of the spatial coordinate system. The plate has dimensions of 0.5 × 0.5 × 0.005 m, an elastic modulus of 200 GPa, and a density of 7.8 × 10⁻⁶ m. 3 kg / m 3 The Poisson's ratio is 0.28. A reconstruction surface is set at a distance of 0.01m from the surface of the plate, with a grid spacing of 0.05m and a size of 0.4×0.4m. Holographic measurement points are uniformly distributed in a 9×9 pattern. A virtual source surface is set at a distance of 0.005m from the surface of the plate, with a wavenumber domain discretization interval of π / 0.05 and a wavenumber domain grid of 27×27. The measurement time is 0.01s, the sampling frequency is 512kHz, and there are 512 sampling points.
[0108] To demonstrate the effectiveness of the VB-AKF reconstruction, two time points were selected to compare the spatial distribution of sound pressure on the reconstruction plane R, at t = 0.98 ms and t = 2.54 ms, corresponding to the 60th and 130th time sampling points, respectively. The results are as follows: Figure 2 As shown.
[0109] To verify the reconstruction effect of the time-domain signal waveform, three points R1(0.1m, 0.1m, 0.01m), R2(-0.1m, 0.05m, 0.01m), and R3(-0.05m, -0.05m, 0.01m) were selected on the reconstruction surface for comparison between the theoretical and reconstructed time-domain sound pressure values. The comparison results are shown in […]. Figure 3 .
[0110] To quantitatively represent the degree of agreement between the reconstruction results and the theoretical results, a phase evaluation factor E is defined. p and amplitude evaluation factor E a The calculation formula is:
[0111]
[0112] In the formula, p t Represents the sound pressure level of the reconstructed surface theory, p r This represents the reconstructed sound pressure level. Where E... p The closer E is to 1, the closer the phase match is. a The closer the value is to 0, the smaller the amplitude error. The results are as follows: Figure 4 As shown, the experimental results further demonstrate that the time-domain sound pressure signal reconstructed using the method of this invention has a good agreement with the theoretical value.
[0113] The above are merely preferred embodiments of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.
Claims
1. A method of transient sound field reconstruction based on a variational Bayesian approximation of an augmented Kalman filter, characterized in that, Includes the following steps: A holographic surface is set in the sound field to be measured, and the time-domain sound pressure signal is measured based on the holographic surface to obtain the measured sound pressure column vector; A virtual source surface is set in the sound field to be measured, and the wavenumber domain is discretized into several wavenumber points based on the virtual source surface; Based on the measured sound pressure column vector and several wavenumber points, the time-domain sound pressure signal is modeled using the time-domain plane wave superposition method to obtain the sound pressure signal model; Based on the sound pressure signal model, an augmented Kalman filter based on variational Bayes approximation is used to jointly recursively estimate the state vector of the sound field and the pressure time wavenumber spectrum on the virtual source surface to obtain the state vector estimate and the wavenumber spectrum estimate. The process of jointly recursively estimating the state vector of the sound field and the pressure time wavenumber spectrum on the virtual source surface using an augmented Kalman filter based on variational Bayesian approximation includes: at the beginning of the time series, initial estimation of the state vector of the sound field and the pressure time wavenumber spectrum on the virtual source surface is performed, and an initial covariance matrix is set; the multi-input multi-output system of the time-domain plane wave superposition method is transformed into a state-space form using a feature system implementation algorithm to obtain the state-space equation and the input-output measurement equation; an augmented state vector is constructed based on the first-order smoothness condition contained in the discrete pressure time wavenumber spectrum on the virtual source surface; the updated state-space equation and the input-output measurement equation are obtained based on the pressure time wavenumber spectrum on the virtual source surface; and a separable approximate distribution is obtained by posterior distribution approximation of the augmented state vector and the updated state-space equation using the standard variational Bayesian method. Based on the state vector estimation and wavenumber spectrum estimation, the sound pressure signal at any spatial point in the sound field at each time moment is reconstructed.
2. The method of claim 1, wherein, The process of modeling time-domain sound pressure signals using the time-domain plane wave superposition method includes: convolution superposition of the time-domain wavenumber spectrum of the virtual source surface and the time-domain propagation kernel; where the expression is: In the formula, Represents temporal convolution. This represents each spatial location within the sound field. It is the wavenumber domain. l wavenumber of each sampling point , ，L The number of sampling points in the wavenumber domain; , and Representing planes respectively R , H and V exist z Position on the axis, where , ; and It is the time-domain sound pressure wavenumber spectrum of the virtual source surface With holographic surface H Time-varying sound pressure and the reconstructed surface R Time-varying sound pressure Associated temporal propagation kernel.
3. The transient sound field reconstruction method based on the variational Bayesian approximation augmented Kalman filter according to claim 2, characterized in that, The expression for the time-domain sound pressure at any point on the holographic surface at any time is: ； in express Time-space holographic surface sound pressure matrix express The wavenumber spectrum matrix of the imaginary source surface sound pressure at time step. express The temporal propagation kernel matrix of the time-domain holographic surface and the virtual source surface.
4. The transient sound field reconstruction method based on the variational Bayesian approximation augmented Kalman filter according to claim 1, characterized in that, The feature system implementation algorithm transforms the multi-input multi-output system of the time-domain plane wave superposition method into a state-space form, yielding the state-space equations and input-output measurement equations. The expressions for these equations are as follows: In the formula, It is the Gaussian process noise matrix. , express The covariance matrix, Represents the measurement noise matrix. , express The covariance matrix, Indicates time step The state matrix, This represents the virtual source surface acoustic pressure wavenumber spectrum matrix. Let A represent the holographic surface sound pressure matrix, and let B, C, and D represent the discretized state, input, output, and feedthrough matrices, respectively.
5. The transient sound field reconstruction method based on the variational Bayesian approximation augmented Kalman filter according to claim 4, characterized in that, The matrix equation corresponding to the first-order smoothness condition contained in the discrete pressure time wavenumber spectrum on the virtual source surface is: In the formula, This represents the wavenumber spectrum of the virtual source surface acoustic pressure.
6. The transient sound field reconstruction method based on the variational Bayesian approximation augmented Kalman filter according to claim 5, characterized in that, The expression for the posterior distribution approximation of the augmented state vector and the updated state-space equation using the standard variational Bayesian method is as follows: In the formula, and They are and Probability density function.