CN111272274A - Closed space low-frequency sound field reproduction method based on microphone random sampling - Google Patents

Closed space low-frequency sound field reproduction method based on microphone random sampling Download PDF

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CN111272274A
CN111272274A CN202010109592.6A CN202010109592A CN111272274A CN 111272274 A CN111272274 A CN 111272274A CN 202010109592 A CN202010109592 A CN 202010109592A CN 111272274 A CN111272274 A CN 111272274A
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sound field
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CN111272274B (en
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陈克安
胥健
王磊
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Northwestern Polytechnical University
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Abstract

The invention provides a reproduction method of a closed space low-frequency sound field based on microphone random sampling, which utilizes a plurality of microphones to sample at random positions in a closed space without utilizing microphone arrays such as a spherical array and the like, ensures the reproduction precision, and has simpler, more convenient and flexible operation and low test cost; then, recording and reproducing the sound field by a space domain decomposition method of the plane wave; aiming at the problem of the calculation speed caused by the large base number of the plane wave model, the method adopts the SALSA (split augmented Lagrange shrinkage Algorithm), and compared with a complex coordinate descent algorithm, the calculation speed is greatly increased, so that the plane wave model can be suitable for the reproduction of a total-space unsteady sound field.

Description

Closed space low-frequency sound field reproduction method based on microphone random sampling
Technical Field
The invention belongs to the technical field of space sound field recording and reproduction, and particularly relates to a microphone random sampling-based closed space low-frequency sound field reproduction method.
Background
In many engineering applications, it is very important to acquire prior information such as spatial sound field distribution characteristics, time-frequency characteristics of field points, and acoustic transfer functions from speakers to microphones (referred to as impulse responses in time domain). For example, when active control is performed on low-frequency noise in a closed space, the first confusion is that the temporal and spatial distribution characteristics of the noise are unknown, so that it is difficult to provide an effective evaluation and solution. Therefore, scholars at home and abroad carry out deep research on the aspect of high-precision restoration of real sound field characteristics, and a plurality of Sound Field Reconstruction (SFR) methods are provided.
The SFR technique is a technique of numerically calculating or predicting a target sound field parameter by using a specific algorithm by creating a mathematical model, and presenting the target sound field in the form of data, an image, audio, or the like (i.e., visualizing or audilizing the sound field). More sophisticated SFR techniques include finite element, boundary element, statistical energy analysis, etc. although these methods are more sophisticated and have commercial software offering powerful computational power, the computational results are still far from the actual sound field distribution due to the difficulty in simulating the real load and boundary conditions. Therefore, the reproduction method based on actual measurement is still the most reliable and efficient method.
Due to the limitation of Shannon-Nyquist sampling theorem, great test cost is consumed for directly measuring the space-time characteristics of the whole sound field point by point in a three-dimensional closed space. An alternative approach is to interpolate and extrapolate the sound field, i.e. estimate its sound pressure at spatially unmeasured locations. The method is based on the basis of the basis function decomposition of the sound field, and the sound pressure of the sound field is reproduced by linear superposition of the product of the basis function and the expansion coefficient of the basis function through solving the expansion coefficient. The method is more suitable for engineering application because the method does not need to predict the geometrical shape and boundary conditions of the closed space and the decomposition form is flexible and various. Common base functions in practice are plane waves, cylindrical harmonics, spherical harmonics, etc.
The SFR method based on the basis function decomposition generally comprises the steps of ① firstly utilizing a microphone array (in various forms such as a circular array, a plane array, a spherical array and the like) to collect a field sound field, ② secondly constructing a least square inverse problem or a sparse recovery problem to solve a basis function (plane wave or spherical harmonic function) expansion coefficient, and ③ later estimating the sound pressure of any field point through a sound field interpolation or extrapolation formula by the expansion coefficient.
In the field of sound field sampling, the spherical array is the first choice due to the advantages of various forms, simple and efficient signal processing method, capability of effectively collecting three-dimensional sound fields from all directions and the like. However, different types of spherical arrays have advantages and disadvantages, for example, a hollow spherical array has a problem of 'Bessel zero point', a rigid spherical array is not suitable for being measured close to the wall surface of a cabin, a heart-shaped spherical array is easy to introduce phase mismatch errors, and the like, so that the spherical array is not good in universality in practical application, and a proper type and configuration need to be selected according to specific conditions. In addition, purchasing a ball array also requires additional capital cost.
In the aspect of a reproduction algorithm, due to the sparsity of the acoustic mode of the closed space, a compressed sensing theory is applied to construct L1The sparse recovery problem of norm relaxation is solved, and the precision of SFR can be ensured while the number of sampling points is reduced. Solving method generally will be L described above1The sparse recovery problem of norm relaxation is equivalent to L in the form of Lagrangian1The norm-constrained least squares problem, also known as the Lasso problem, is solved using a complex coordinate descent algorithm.
In addition, the basis functions are typically selected to be plane waves or spherical harmonics. The spherical harmonic model has small dimension and high calculation speed, generally adopts a spherical array, can effectively reproduce a sound field in an area near the spherical array, but can generate sound field distortion at a position far away from the spherical array, so the method is more suitable for SFR (small frequency response) of local space steady state and unsteady state; the plane wave model can adopt various array forms, is suitable for steady state SFR of a full space, but has large dimension and slow calculation speed.
Disclosure of Invention
In order to save the acquisition cost of spherical array equipment and simplify the optimization selection work of spherical array types and configuration, the invention provides the method for sampling at random positions in a closed space by using a plurality of microphones, so that the operation is simpler, more convenient and more flexible, and sound field recording and reproduction are carried out by a space domain decomposition method of plane waves.
In addition, aiming at the problem of the calculation speed caused by the large base number of the plane wave model, the invention adopts a Split Augmented Lagrange Shrinkage Algorithm (SALSA) and has a calculation speed which is greatly accelerated compared with a complex coordinate reduction algorithm, so that the plane wave model can be suitable for the reproduction of a total-space unsteady sound field.
The technical scheme of the invention is as follows:
the method for reproducing the low-frequency sound field in the closed space based on the random sampling of the microphone is characterized by comprising the following steps of: the method comprises the following steps:
step 1: establishing a rectangular coordinate system in a closed space V;
step 2: in the rectangular coordinate system established in the step 1, Q measuring point positions are randomly selected, a microphone is placed, and the position vector of the Q measuring point is recorded as rq＝[xq,yq,zq]T，q＝1,2,...,Q，(·)TRepresenting a transpose; sampling a sound field in a closed space V by using the Q microphones to obtain complex sound pressure vectors at the positions of the Q microphones under the frequency f
Figure BDA0002389510230000031
Wherein
Figure BDA0002389510230000032
Representing a complex set;
and step 3: constructing a plane wave spatial domain transform matrix
Figure BDA0002389510230000033
Figure BDA0002389510230000034
In the formula sn＝[sinθncosφn,sinθnsinφn,cosθn]TDenotes the propagation direction of the nth plane wave, N being 1,2nIs the angle of orientation, θnAt a pitch angle of αnIs a set plane wave discrete correlation constant, e represents a natural base number,
Figure BDA0002389510230000035
represents an imaginary unit; k is 2 pi f/c is wave number, c is sound velocity;
and 4, step 4: according to
Figure BDA0002389510230000036
The Lasso problem is constructed, wherein
Figure BDA0002389510230000037
Is a complex amplitude vector of the plane wave to be solved, and N represents the number of discrete plane waves; lambda belongs to (0, | | H)Hp||∞]Is a sparsity adjustment parameter (.)HRepresenting the conjugate transpose, | · | | non-conducting phosphor∞Represents infinite norm, | ·| non-conducting phosphor2Represents L2Norm, | · | luminance1Represents L1A norm;
and 5: solving the Lasso problem constructed in the step 4 by using SALSA to obtain a plane wave complex amplitude vector w;
step 6: by using
Figure BDA0002389510230000038
At an estimated frequency f, an arbitrary field point rmAcoustic pressure of
Figure BDA0002389510230000039
Wherein
Figure BDA00023895102300000310
Further, in step 4, the larger the value of the sparsity adjusting parameter λ is, the more sparse or compressible w is.
Further, in step 4, the sparsity adjusting parameter λ is determined by a cross validation method, and the specific process is as follows:
step 4.1: arbitrary partitioning of a vector p into NgNon-overlapping subsets to obtain
Figure BDA00023895102300000311
p(g)The g-th subset representing p;
step 4.2: in the interval (0, lambda)nom]To uniformly select NλValues as possible values of λ:
Figure BDA00023895102300000312
wherein λnomIs a set preselected value upper limit;
step 4.3: according to
Figure BDA00023895102300000313
Calculating the error of the g subset
Figure BDA00023895102300000314
Wherein H(g)Is the g-th subset p in H(g)A matrix of corresponding columns; w is a(-g)Is λ ═ λnIn time, the Lasso problem is solved by using SALSA
Figure BDA0002389510230000041
The solution obtained, p(-g)Indicates p deletion p(g)Vector obtained after neutralization of element, H(-g)Indicates H deletes H(g)The matrix obtained after the column(s) in (1);
step 4.4: according to
Figure BDA0002389510230000042
Calculating to obtain lambdanCorresponding error En；
Step 4.5: according to λ ═ λn′Determining a final parameter λ, wherein
Figure BDA0002389510230000043
Further, the specific process for solving the Lasso problem by using the SALSA is as follows:
step 5.1: initializing parameters, and enabling w to be 0, d to be 0, v to be 0 and mu to be more than 0;
step 5.2: are respectively in accordance with
Figure BDA0002389510230000044
v＝soft(w-d,λ/μ)+d
d＝v-w
Updating w, v, and d, wherein soft (·,) represents a soft threshold function;
step 5.3: if the termination condition is reached: maximum number of iterations or threshold ΔwE, w is the final plane waveA complex amplitude vector; otherwise, repeating the step 5.2; deltawIs | | | w(i)-w(i-1)||∞/||w(i)||∞I denotes the current number of iterations, ∈
Is a set, very small positive number.
Further, in step 4.1, NgThe value is 4 or 5.
Further, in step 4.2, λnomTake | | | HHp||∞/50，NλThe value range of (A) is 10-50.
Further, in step 3, the direction angle phinE is 0,360 degree, pitch angle thetan∈[0,180°]。
Further, in step 3, the determination of N directions is obtained by approximately uniformly sampling a spherical surface based on the Thomson problem.
Further, in step 3, α is taken1＝α2＝…＝αN＝4π/N。
Advantageous effects
(1) The invention adopts the microphones which are randomly distributed in the closed space to sample the field sound field, does not need to utilize the microphone arrays such as a spherical array and the like, ensures the reproduction precision, has simpler and more flexible operation and low test cost.
(2) The invention solves the Lasso problem of plane wave decomposition by using SALSA, greatly accelerates the calculation speed, and ensures that the plane wave model can be suitable for the reproduction of unsteady sound fields.
Additional aspects and advantages of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention.
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The above and/or additional aspects and advantages of the present invention will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
FIG. 1 is a flow chart of a method for reproducing a low-frequency sound field in a closed space based on microphone random sampling according to the invention;
FIG. 2 is a schematic diagram of a desired sound field in a rectangular enclosure with rigid walls;
FIG. 3 is a schematic diagram of a randomly sampled microphone distribution within an enclosed space;
FIG. 4 is a schematic representation of a sound field reproduced using the method of the present invention;
fig. 5 is a schematic diagram of the spatial relative error of a reproduced sound field using the method of the present invention.
Detailed Description
The invention provides a method for sampling at random positions in a closed space by using a plurality of microphones, which is simpler and more flexible to operate, and records and reproduces a sound field by a space domain decomposition method of plane waves; aiming at the problem of the calculation speed caused by the large base number of the plane wave model, a Split Augmented Lagrangian Shrinkage Algorithm (SALSA) is adopted, and compared with a complex coordinate reduction algorithm, the calculation speed is greatly increased, so that the plane wave model can be suitable for the reproduction of a total-space unsteady sound field.
The method comprises the following specific steps:
step 1: in a specific closed space V, a rectangular coordinate system is established according to actual conditions.
Step 2: in the established rectangular coordinate system, Q measuring point positions are randomly selected, a microphone is placed, and the position vector of the Q measuring point is recorded as rq＝[xq,yq,zq]T(Q ═ 1,2,.., Q), where (·)TIndicating transposition. Sampling a sound field in a closed space V by using the Q microphones to obtain complex sound pressure vectors at the positions of the Q microphones under the frequency f
Figure BDA0002389510230000051
Where k is 2 pi f/c is the wave number, where c is the speed of sound,
Figure BDA0002389510230000052
a complex set is represented. (frequency (k) is omitted for simplicity).
And step 3: constructing a 'plane wave space domain transformation matrix'
Figure BDA0002389510230000053
The specific elements are as follows
Figure BDA0002389510230000061
In the formula, sn＝[sinθncosφn,sinθnsinφn,cosθn]T(N ═ 1, 2.., N) denotes the propagation direction (Φ) of the nth plane waven,θn) Wherein phinE [0,360 deg. ] is direction angle thetan∈[0,180°]Determination of N directions for pitch angle by approximately uniform sampling of the sphere based on Thomson's problem αn(N1, 2.., N) is a constant related to plane wave dispersion, and for the approximately uniform dispersion strategy employed by the present invention, there is approximately α1＝α2＝…＝αN4 pi/N; e represents a natural base number of the image,
Figure BDA0002389510230000062
representing imaginary units.
And 4, step 4: according to
Figure BDA0002389510230000063
The Lasso problem is constructed, wherein
Figure BDA0002389510230000064
Is a complex amplitude vector of the plane wave to be solved, and N represents the number of discrete plane waves; lambda belongs to (0, | | H)Hp||∞]Is a sparsity adjustment parameter (.)HRepresenting the conjugate transpose, | · | | non-conducting phosphor∞Representing an infinite norm, the larger the value of λ, the more sparse or compressible w; i | · | purple wind2Represents L2Norm, | · | luminance1Represents L1And (4) norm.
And 5: and solving the Lasso problem constructed in the fourth step by using the SALSA to obtain a plane wave complex amplitude vector w.
Step 6: by using
Figure BDA0002389510230000065
At an estimated frequency f, an arbitrary field point rmOfSound pressure
Figure BDA0002389510230000066
Wherein
Figure BDA0002389510230000067
The sparsity adjusting parameter lambda can be determined by adopting a cross validation method, and the specific process is as follows:
step 4.1: arbitrary partitioning of a vector p into NgNon-overlapping subsets, i.e.
Figure BDA0002389510230000068
Where N isgThe value can be 4 or 5;
step 4.2: in the interval (0, lambda)nom]To uniformly select NλThe value is taken as a possible value of λ, i.e.
Figure BDA0002389510230000069
Wherein λnomSuggested as | | HHp||∞/50，NλThe value range of (1) is 10-50;
step 4.3: according to
Figure BDA00023895102300000610
Calculating the error of the g subset
Figure BDA00023895102300000611
Wherein p is(g)Denotes the g-th subset of p, H(g)Is the g-th subset p in H(g)Matrix of corresponding columns, p(-g)Indicates p deletion p(g)Vector obtained after neutralization of element, H(-g)Indicates H deletes H(g)The matrix obtained after the column(s) in (1); w is a(-g)Is λ ═ λnIn time, the Lasso problem is solved by using SALSA
Figure BDA0002389510230000071
The solution obtained;
step 4.4: according to
Figure BDA0002389510230000072
Calculating to obtain lambdanCorresponding error En；
Step 4.5: according to λ ═ λn′Determining a final parameter λ, wherein
Figure BDA0002389510230000073
The method for solving the Lasso problem by using the SALSA
Figure BDA0002389510230000074
The specific process is as follows:
step 5.1: initializing parameters, and enabling w to be 0, d to be 0, v to be 0 and mu to be more than 0 (an empirical value is lambda/10);
step 5.2: are respectively in accordance with
Figure BDA0002389510230000075
v＝soft(w-d,λ/μ)+d
d＝v-w
Updating w, v, and d, wherein soft (·,) represents a soft threshold function;
step 5.3: if the end condition is reached, the maximum number of iterations I or the threshold value DeltawIf not more than epsilon, w is the final plane wave complex amplitude vector; otherwise, repeating the step 5.2; the value range of I is 500-1000; deltawIs | | | w(i)-w(i-1)||∞/||w(i)||∞I represents the current number of iterations, epsilon is a small positive number, and can be taken to be 10-4。
The present invention will be further described with reference to the following drawings and simulation examples, which include, but are not limited to, the following examples.
The basic flow of the method for reproducing the low-frequency sound field of the closed space based on the random sampling of the microphone is shown in figure 1, and the specific process is as follows:
1. given a rectangular enclosure of rigid walls with dimensions 3.7m × 1.8m × 1.2m, the origin of coordinates is chosen to be a few in the enclosureAnd a center of the line. A monopole point source is applied to the position (1.83,0.88,0.58) with the intensity of 0.0001+0.0001i m3And/s, producing a desired sound field to be reproduced. The frequencies under examination of the examples were chosen to be 100Hz and 195Hz, which are the non-resonant frequency and the approximate resonant frequency of the enclosed space, respectively. Fig. 2 shows the expected sound field of the closed space at 100Hz and 195Hz, respectively, by plotting the real part of sound pressure at the field point. The field points are 8000 observation points which are evenly distributed in space.
2. In a closed space, the Q is randomly distributed to 36 measuring points, and microphones are placed, as shown in figure 3. An expected sound field vector consisting of sound pressure at the position of a sampling point is obtained by sampling the expected sound field
Figure BDA0002389510230000081
3. Approximately and uniformly sampling the spherical surface based on Thomson problem, wherein the number N of sampling points is 400, and the direction (phi) of the sampling pointsn,θn) The (n is 1 to 400) is the direction of the discrete plane wave. Obtaining a 'plane wave space domain transformation matrix' according to the following formula "
Figure BDA0002389510230000082
Figure BDA0002389510230000083
Wherein e represents a natural base number,
Figure BDA0002389510230000084
denotes an imaginary unit, k 2 pi f/c is a wave number, and c is a sound velocity, (. DEG)TRepresenting a transpose; sn＝[sinθncosφn,sinθnsinφn,cosθn]T(n is 1 to 400) represents the propagation direction (phi) of the nth plane waven,θn) Wherein phinE [0,360 deg. ] is direction angle thetan∈[0,180°]Is a pitch angle.
4. The structural Lasso problem is shown by the following formula
Figure BDA0002389510230000085
Wherein
Figure BDA0002389510230000086
Is a complex amplitude vector of plane wave to be solved, | · | | luminance2Represents L2Norm, | · | luminance1Represents L1And (4) norm. Lambda > 0 is a sparsity adjusting parameter which directly controls the sparsity of the solution w, the larger the lambda value is, the more sparse the solution is, but when lambda is larger than HHp||∞When the temperature of the water is higher than the set temperature,
since the calculation result w is 0, it is meaningless, and therefore, it may be in the interval (0, | H)Hp||∞/50]A suitable lambda value is determined. The invention adopts a cross-validation method to determine the parameter lambda, namely dividing the vector p into Ng5 non-overlapping subsets in the interval (0, | HHp||∞/50]To uniformly select Nλ20 possible values
Figure BDA0002389510230000087
According to
Figure BDA0002389510230000088
Calculating the error of the g subset
Figure BDA0002389510230000089
Wherein, w(-g)Is λ ═ λnSolving the Lasso problem by using the SALSA
Figure BDA00023895102300000810
The solution obtained; p is a radical of(g)Denotes the g-th subset of p, H(g)Is the g-th subset p in H(g)Matrix of corresponding columns, p(-g)Indicates p deletion p(g)Vector obtained after neutralization of element, H(-g)Indicates H deletes H(g)The matrix obtained after the column(s) in (1); then according to
Figure BDA00023895102300000811
Calculating to obtain lambdanCorresponding error En(ii) a Finally, theAccording to λ ═ λn′Determining a final parameter λ, wherein
Figure BDA00023895102300000812
In this embodiment, the parameters λ are determined to be 1.24 × 10 by cross-validation-4(100Hz) and 0.004(195 Hz).
5. Solving the Lasso problem using the SALSA
Figure BDA0002389510230000091
Obtaining a complex amplitude w of the plane wave, which comprises the following specific steps:
(1) initializing parameters, and enabling w to be 0, d to be 0, v to be 0 and mu to be lambda/10;
(2) are respectively in accordance with
Figure BDA0002389510230000092
v＝soft(w-d,λ/μ)+d
d＝v-w
Updating w, v, and d, wherein soft (·,) represents a soft threshold function;
(3) if the end condition is reached, the maximum number of iterations I or the threshold value DeltawIf not more than epsilon, w is the final plane wave complex amplitude vector; otherwise, repeating the step (2); in this embodiment, I is 1000; deltawIs | | | w(i)-w(i-1)||∞/||w(i)||∞I denotes the current number of iterations, ε is taken to be 10-4。
6. By using
Figure BDA0002389510230000093
Estimating an arbitrary field point rmAcoustic pressure of
Figure BDA0002389510230000094
Wherein
Figure BDA0002389510230000095
As shown in fig. 4, the sound field is reproduced in full space at 100Hz and 195Hz respectively (estimated real part of sound pressure at 8000 field points). As can be seen from the observation, the utilization of the air is36 microphones which are randomly distributed in the middle are sampled, and the distribution of a full-space sound field can be well reproduced. To further quantify the accuracy of the evaluation SFR, the evaluation index at frequency f is given as follows:
① field point relative error er
Figure BDA0002389510230000096
② Total spatial average relative error Er
Figure BDA0002389510230000097
Wherein M is the number of the field points.
③ mode assurance criterion MAC
Figure BDA0002389510230000101
Wherein the content of the first and second substances,
Figure BDA0002389510230000102
to evaluate the desired sound pressure vector at the site,
Figure BDA0002389510230000103
to evaluate the reproduced sound pressure vector at the site. The MAC value range is usually 0-1, and the closer to 1, the more similar the spatial distribution of the reproduced sound field and the expected sound field is.
FIG. 5 shows the relative error for 8000 field points at 100Hz and 195Hz, and it can be seen that the majority of the relative errors are below 20%. Full space average relative error E at two frequenciesrAnd the mode assurance criterion MAC is respectively 7.33 percent and 0.9947(100Hz) and 7.91 percent and 0.9941(195Hz), which shows that the method provided by the invention can more accurately reproduce the sound pressure and the full-space sound field distribution of the unmeasured field points.
Table 1 compares the CPU time required for plane wave decomposition using the complex coordinate descent algorithm and the SALSA when the number Q of random sampling points is 36 and the number N of discrete plane waves is 200, 400, and 700, respectively. It can be seen that the SALSA adopted by the invention greatly improves the calculation speed of plane wave decomposition and is convenient for data processing and analysis. The method is suitable for reproduction of a total-space unsteady sound field when the number N of the discrete plane waves is small.
TABLE 1 plane wave decomposition computation time
Figure BDA0002389510230000104
Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention, and that variations, modifications, substitutions and alterations can be made in the above embodiments by those of ordinary skill in the art without departing from the principle and spirit of the present invention.

Claims (9)

1. A reproduction method of a low-frequency sound field in a closed space based on microphone random sampling is characterized by comprising the following steps: the method comprises the following steps:
step 1: establishing a rectangular coordinate system in a closed space V;
step 2: in the rectangular coordinate system established in the step 1, Q measuring point positions are randomly selected, a microphone is placed, and the position vector of the Q measuring point is recorded as rq＝[xq,yq,zq]T，q＝1,2,...,Q，(·)TRepresenting a transpose; sampling a sound field in a closed space V by using the Q microphones to obtain complex sound pressure vectors at the positions of the Q microphones under the frequency f
Figure FDA0002389510220000011
Wherein
Figure FDA0002389510220000012
Representing a complex set;
and step 3: constructing a plane wave spatial domain transform matrix
Figure FDA0002389510220000013
Figure FDA0002389510220000014
In the formula sn＝[sinθncosφn,sinθnsinφn,cosθn]TDenotes the propagation direction of the nth plane wave, N being 1,2nIs the angle of orientation, θnAt a pitch angle of αnIs a set plane wave discrete correlation constant, e represents a natural base number,
Figure FDA0002389510220000015
represents an imaginary unit; k is 2 pi f/c is wave number, c is sound velocity;
and 4, step 4: according to
Figure FDA0002389510220000016
The Lasso problem is constructed, wherein
Figure FDA0002389510220000017
Is a complex amplitude vector of the plane wave to be solved, and N represents the number of discrete plane waves; lambda belongs to (0, | | H)Hp||∞]Is a sparsity adjustment parameter (.)HRepresenting the conjugate transpose, | · | | non-conducting phosphor∞Represents infinite norm, | ·| non-conducting phosphor2Represents L2Norm, | · | luminance1Represents L1A norm;
and 5: solving the Lasso problem constructed in the step 4 by using SALSA to obtain a plane wave complex amplitude vector w;
step 6: by using
Figure FDA0002389510220000018
At an estimated frequency f, an arbitrary field point rmAcoustic pressure of
Figure FDA0002389510220000019
Wherein
Figure FDA00023895102200000110
2. The method for reproducing the low-frequency sound field in the closed space based on the random sampling of the microphone as claimed in claim 1, wherein: in step 4, the larger the value of the sparsity adjusting parameter lambda is, the more sparse or compressible w is.
3. A reproduction method of a closed space low-frequency sound field based on microphone random sampling according to claim 1 or 2, characterized in that: in the step 4, the sparsity adjusting parameter lambda is determined by adopting a cross validation method, and the specific process is as follows:
step 4.1: arbitrary partitioning of a vector p into NgNon-overlapping subsets to obtain
Figure FDA0002389510220000021
p(g)The g-th subset representing p;
step 4.2: in the interval (0, lambda)nom]To uniformly select NλValues as possible values of λ:
Figure FDA0002389510220000022
wherein λnomIs a set preselected value upper limit;
step 4.3: according to
Figure FDA0002389510220000023
Calculating the error of the g subset
Figure FDA0002389510220000024
Wherein H(g)Is the g-th subset p in H(g)A matrix of corresponding columns; w is a(-g)Is λ ═ λnIn time, the Lasso problem is solved by using SALSA
Figure FDA0002389510220000025
The solution obtained, p(-g)Indicates p deletion p(g)After the element is neutralized to obtainVector of (a), H(-g)Indicates H deletes H(g)The matrix obtained after the column(s) in (1);
step 4.4: according to
Figure FDA0002389510220000026
Calculating to obtain lambdanCorresponding error En；
Step 4.5: according to λ ═ λn′Determining a final parameter λ, wherein
Figure FDA0002389510220000027
4. The method for reproducing the low-frequency sound field in the closed space based on the random sampling of the microphone as claimed in claim 1, wherein: the specific process for solving the Lasso problem by using the SALSA is as follows:
step 5.1: initializing parameters, and enabling w to be 0, d to be 0, v to be 0 and mu to be more than 0;
step 5.2: are respectively in accordance with
Figure FDA0002389510220000028
v＝soft(w-d,λ/μ)+d
d＝v-w
Updating w, v, and d, wherein soft (·,) represents a soft threshold function;
step 5.3: if the termination condition is reached: maximum number of iterations or threshold ΔwIf not more than epsilon, w is the final plane wave complex amplitude vector; otherwise, repeating the step 5.2; deltawIs | | | w(i)-w(i-1)||∞/||w(i)||∞I denotes the current number of iterations and epsilon is a set small positive number.
5. The method for reproducing the low-frequency sound field in the closed space based on the random sampling of the microphone as claimed in claim 3, wherein: in step 4.1, NgThe value is 4 or 5.
6. The method for reproducing the low-frequency sound field in the closed space based on the random sampling of the microphone as claimed in claim 3, wherein: in step 4.2, λnomTake | | | HHp||∞/50，NλThe value range of (A) is 10-50.
7. The method for reproducing the low-frequency sound field in the closed space based on the random sampling of the microphone as claimed in claim 1, wherein: in step 3, the direction angle phinE is 0,360 degree, pitch angle thetan∈[0,180°]。
8. The method for reproducing the low-frequency sound field in the closed space based on the random sampling of the microphone as claimed in claim 1, wherein: in step 3, the determination of the N directions is obtained by approximately uniform sampling of a spherical surface based on the Thomson problem.
9. The method for reproducing the low-frequency sound field in the closed space based on the random sampling of the microphone as claimed in claim 1, wherein in the step 3, α is taken1＝α2＝…＝αN＝4π/N。
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