A method for optimizing a multi-target beamformer in a spherical harmonic domain

Abstract

The application relates to a multi-target beam optimization problem in a spherical harmonic domain, and specifically provides a multi-target beam optimization method in a spherical harmonic domain based on a multi-target intelligent optimization algorithm. The method uses the characteristics that the expression of a white noise gain and a directivity factor of a beamformer in a spherical harmonic domain is a positive definite quadratic form, sets thresholds for the white noise gain and the directivity factor in advance, and determines the range of beamforming weights to be optimized based on the thresholds; the beam optimization problem in the spherical harmonic domain is converted into a multi-target optimization problem; finally, the NSGA-Ⅱ algorithm with constraint processing is used to solve the problem, a set of beamforming weights satisfying Pareto optimization of each target is obtained, and the design of an optimal beamformer in the spherical harmonic domain is realized. The method can make the overall sidelobe level, the directivity factor and the white noise gain of the beamformer satisfy the beamforming weights of the Pareto optimization, and there is no complex parameter adjustment process.

Description

Technical Field

[0001] This invention relates to beamformers, specifically to a multi-objective beam optimization problem in the spherical harmonic domain, and provides a method for optimizing multi-objective beamformers in the spherical harmonic domain based on a multi-objective intelligent optimization algorithm. Background Technology

[0002] In recent years, microphone arrays have been widely used in scenarios such as smartphones, teleconferencing systems, indoor and outdoor noise monitoring, and indoor acoustic analysis for noise and echo suppression, dereverberation, single / multiple sound source localization, and multi-speech source separation.

[0003] Microphone arrays come in various structures, including linear arrays, rectangular arrays, circular arrays, and spherical arrays. Compared to other microphone array types, spherical arrays offer several advantages: the unique rotational symmetry of the beam pointing of a spherical microphone array facilitates beamforming and spatial filtering, allowing for effective enhancement or suppression of target sound sources in any direction; analytical representation in the spherical harmonic domain effectively reduces computational complexity; and spherical arrays possess the advantage of decoupling frequency components from angle-dependent spherical harmonic functions in the spherical harmonic domain, making it easy to design broadband beamformers.

[0004] When strong random interference signals exist in the array, the adaptive null steering algorithm struggles to converge quickly and effectively nullify the interference signals, leading to a sharp decline in array beamforming performance. In such cases, it is desirable to design a fixed beamformer that possesses both low sidelobe levels, good directivity, and high robustness.

[0005] Current traditional single-objective beamformer design methods struggle to balance performance across various metrics, including sidelobe suppression, directivity factor, and white noise gain. While there are methods that transform the optimization problem of beam patterns in the spherical harmonic domain into a multi-constraint single-objective optimization problem—such as beamformer design methods that maximize array directivity under constraints of white noise gain and sidelobe level—these methods essentially still optimize for a single objective. Therefore, the resulting beamformer's performance may not achieve overall optimization. Furthermore, the selection of constraint values ​​is also problematic, requiring a high level of theoretical knowledge and engineering experience from the user. Summary of the Invention

[0006] To address the challenges of rapid convergence in adaptive beamformers, which hinders their ability to effectively zero out randomly occurring interference signals, and the limitations of existing fixed beamformer design methods in the spherical harmonic domain (SHNDMA) that cannot simultaneously consider multiple design objectives related to array characteristics, this invention proposes a multi-objective beamformer optimization method for spherical microphone arrays based on the NSGA-II with constraint processing. This method rewrites the beamforming pattern optimization problem in the SHNDMA as a constrained multi-objective optimization problem by providing the beamforming expression and determining the beamforming weights. By using the beamforming weights as optimization variables in the multi-objective optimization problem, a Pareto optimal beamforming weight is designed, thereby achieving the optimization of the optimal beamformer in the SHNDMA.

[0007] This invention discloses a method for optimizing multi-target beamformers in the spherical harmonic domain, comprising the following steps:

[0008] S1. The three performance indicators of the beamformer—white noise gain, directivity factor, and overall sidelobe level—are formalized and expressed as three objective functions of a multi-objective optimization problem.

[0009] S2. Treat the distortion-free constraint of the beamformer array on the desired direction as an equality constraint in a multi-objective optimization problem;

[0010] S3. Set constraint values ​​for white noise gain and directionality factor. Combining the characteristics of the positive definite quadratic form of the expression for white noise gain and directionality factor, determine the range of beamforming weights and use them as the search space of the variables to be optimized in the multi-objective optimization problem.

[0011] S4. Finally, the NSGA-II algorithm with constraints is used to solve the proposed multi-objective optimization problem, obtain the optimal beamforming weights, and realize the optimization of the optimal beamformer in the spherical harmonic domain.

[0012] The expression for axisymmetric beamforming in the spherical harmonic domain is:

[0013]

[0014] The above equation can be written in matrix form as follows:

[0015] y(Θ)=d T v(Θ) (2)

[0016] In the above formula, the superscript T indicates that the vector or matrix is ​​transposed, and the (N+1)*1 dimensional vector d is given by the following formula:

[0017] d = [d0, d1, ..., d N ] T (3)

[0018] The (N+1)*1 dimensional vector v(Θ) is given by the following equation:

[0019]

[0020] In the above formula, Θ represents the angle between the desired direction of the array and the direction of arrival of the plane wave, N represents the beamforming order, and P n (.) denotes an nth-order associated Legendre polynomial, and we can see that the array's beam pattern is only affected by d. n With the control of (n=0,1,…,N), we can determine that the variable to be optimized in the subsequent algorithm is d.

[0021] The beamforming expression gives three important measures related to beamformer performance:

[0022] The first metric is white noise gain, defined as the ratio of the signal-to-noise ratio at the array output to the signal-to-noise ratio at the array input. A higher white noise gain in the beamformer indicates better white noise immunity. The expression for white noise gain is:

[0023]

[0024] In the above formula, d is the beamforming weight vector in the spherical harmonic domain, and d T This indicates the transpose of d;

[0025] Matrix A is:

[0026] A = vv T (6)

[0027] Where v T This represents the transpose of vector v, where vector v is:

[0028] v = v(0) = [1, 3, ..., (2N+1)] T (7)

[0029] Matrix B is:

[0030]

[0031] Where M is the number of microphones in the array, diag(.) means constructing a diagonal matrix from the vectors in parentheses, b n (n = 0, ..., N) represents the modal intensities related to the array configuration. Common array configurations include rigid spheres and open spheres, and the corresponding modal intensities are shown below:

[0032]

[0033] Where j n and These are the nth-order spherical Bessel function and the nth-order spherical Hankel function of the second kind, respectively, j' n and Let i and k represent the first derivatives of the corresponding functions, i be the imaginary unit, k be the wave number, and a be the radius of the sphere array.

[0034] The second metric is the directivity factor, which is the ratio of the peak value to the mean value of the squared beam pattern. The larger the directivity factor of the beamformer, the better its directivity. The expression for the directivity factor is:

[0035]

[0036] In the formula, matrix C is:

[0037]

[0038] The third metric is the sum of sidelobe levels (SLS), which is the sum of sidelobe amplitudes at all discrete points after discrete sampling of the sidelobe region. The lower the sum of sidelobe levels of the beamformer, the better its suppression effect on interference signals from outside the desired direction of the array. The expression for the sum of sidelobe levels is as follows:

[0039]

[0040] In the formula Ω SL Represents the sidelobe range, I represents the total number of discrete sampling points in the sidelobe region, and θ i ,φ i Let represent the elevation angle and azimuth angle at the i-th sampling point, respectively.

[0041] Furthermore, S2 provides an equality constraint on the beamforming vector, which constrains the array to have a distortion-free response to signals from the desired direction. The expression for the distortion-free response constraint is:

[0042] h(d)=d T v(0)=1 (13).

[0043] Furthermore, S3 determines the range of beamforming weights. First, a minimum threshold is set for both the white noise gain and the directivity factor. Without loss of generality, these minimum thresholds are denoted as ε. b and ε a Combining equations (5), (10), and (13), we can obtain the following equation:

[0044]

[0045]

[0046] Then, simplifying and decomposing the above two equations, we get the following equation:

[0047]

[0048]

[0049] Finally, the range of values ​​for the beamforming weights under the constraints of white noise gain and directivity factor is obtained as follows:

[0050]

[0051] l = -h (19)

[0052] In the above formula, vectors l and h represent the lower and upper bounds of the obtained beamforming weights, respectively, and min(.) represents the minimum value operation, which is achieved by adjusting ε. a and ε b The value of can control the range of the beamforming weight vector d to be optimized, enabling the algorithm to converge to the optimal solution more quickly.

[0053] Furthermore, the multi-objective optimization problem described in S4 is the beam optimization problem in the spherical harmonic domain, which is written as the following multi-objective optimization problem:

[0054]

[0055] Although the aforementioned constrained multi-objective optimization problem does not have a closed-form solution, it can be solved using intelligent optimization algorithms. This invention uses the NSGA-II algorithm with constraint processing to solve the above problem.

[0056] The NSGA-II algorithm mainly consists of two parts: non-dominated sorting and crowding calculation. Non-dominated sorting is a sorting method used to rank individuals in the population to distinguish different levels of the Pareto front. Crowding calculation is a technique to ensure the uniform distribution of the Pareto front.

[0057] The specific steps of the NSGA-II algorithm are as follows:

[0058] 1) Initialize the population and generate an initial set of individuals;

[0059] 2) Evaluate each individual and calculate the objective function value;

[0060] 3) Perform non-dominated sorting to determine the different levels of the Pareto front;

[0061] 4) Calculate the crowding degree of each individual, and select individuals to generate offspring according to the non-dominated order and the crowding degree calculation results;

[0062] 5) New individuals are generated through crossover and mutation, replacing the original individuals;

[0063] 6) Repeat steps 2)-5) until the stopping criterion is met.

[0064] The main advantages of the NSGA-II algorithm are that it can handle multiple objective functions simultaneously and find a set of optimal solutions in the Pareto front, and that the algorithm converges quickly and is highly efficient. Furthermore, the NSGA-II algorithm introduces a crowding calculation method, which ensures a uniform distribution of the Pareto front, thereby avoiding excessive clustering of individuals in the same region and reducing the risk of finding local optima.

[0065] The original NSGA-II algorithm is only suitable for solving ordinary multi-objective optimization problems. In ordinary multi-objective optimization problems, the objective function usually has no constraints and can be solved by directly calculating the Pareto front. However, in some applications, the solution to the optimization problem may be limited by certain constraints. These constraints are usually equality or inequality equations that define the set of feasible solutions. In this case, it is necessary to integrate these constraints into the multi-objective optimization algorithm to ensure that the generated solutions are all feasible.

[0066] The NSGA-II algorithm with constraint handling addresses this problem by using constraint handling methods. The core idea of ​​this algorithm is to transform constraints into penalty functions and then add these penalty functions to the original objective function. The penalty function's role is to disadvantage infeasible solutions in the search space, thereby increasing the likelihood of generating feasible solutions. Unlike traditional penalty function methods, this algorithm uses an adaptive mechanism to adjust the parameters of the penalty function to ensure that infeasible solutions are not excessively penalized during the search process.

[0067] Furthermore, the solution of the multi-objective optimization problem described in S4 includes the following steps:

[0068] Step 1: Set parameters:

[0069] Set the range of the side lobe Ω SL The number of discrete sampling points I for the sidelobe level, the configuration of the spherical microphone array (including the number of microphones, array radius, and other parameters), the frequency point f for beamforming, and the beamforming order N. order White noise gain threshold ε b Directionality factor threshold ε a Parameters related to the NSGA-II algorithm: including population size N pop Maximum number of generations N max Optimize the number of variables v to simulate the crossover exponent η of binary crossover. c and the variation exponent η used for polynomial variation m Crossover probability p m (A value between 0 and 1 controls the probability of an individual mutating);

[0070] The second step is to initialize the population:

[0071] Initialize the first generation population by randomly generating N within the range defined by l and h. pop The instance d corresponding to the beamforming weight vector d n (n=1,2,...N, pop (i.e., generating N) pop Replace d with a real number. n d in n (Instances of n = 0, ..., N), and these constitute the first generation population, where each beam in the population forms a weight vector instance d. n Also known as an individual in the population, set the current population generation N. c =1;

[0072] The third step is to calculate the modified objective function value:

[0073] For each individual d in the current population n Constraint handling techniques are applied to each objective function dimension i to compute the modified objective function value F. i (d n );

[0074] Step 4: Calculate the Pareto level and congestion distance:

[0075] Using F i (d n For each individual d in the current population n Perform a non-dominated sort and assign Pareto levels to them. Then, in each Pareto level set, use F... i (d n ) Calculate d for each individual n Crowded distance;

[0076] Step 5: Selection of paternal genes.

[0077] Select N using binary tournament selection method for the current population. pop Among the individuals, the fitness comparison criteria are as follows: individuals with smaller Pareto scores have higher fitness; when Pareto scores are the same, individuals with greater crowding distances have higher fitness; and finally, individuals with higher overall fitness are selected.

[0078] Step 6: Producing offspring individuals:

[0079] Each time, without replacement, two individuals are randomly selected from the parent generation as the parent and the target, respectively. Simulated binary crossover and polynomial mutation are then performed on these individuals to produce offspring. N pop The parent individuals produced a total of N. pop Individual offspring;

[0080] Step 7: Update the current population:

[0081] The set of offspring individuals generated in step six and the current population are combined to form a new population of size 2N. pop Given a set, perform the operation in step four on that set, and then select N based on fitness. pop Individuals replace the current population to form a new generation of population, for generations N. c Add one;

[0082] Step 8: Judgment

[0083] Determine the current algebra N c Is it greater than the maximum number of generations N? max If it is greater than , skip to step nine; otherwise, skip to step three.

[0084] Ninth step output:

[0085] Output the set of feasible individuals in the current population. Each individual in the set is a beamforming weight that satisfies Pareto optimality in all measures.

[0086] This invention provides a solution for multi-target beamforming in the spherical harmonic domain. The method simultaneously optimizes the white noise gain, directivity factor, and sidelobe level of the beamformer to maintain a low overall sidelobe level in spherical harmonic beamforming, while controlling other beamformer performance metrics. Furthermore, the method allows for control over the optimization range. This approach solves the problem of excessively high overall sidelobe levels in traditional spherical harmonic beamforming, avoids frequent constraint parameter settings, and yields multiple beamformers with different Pareto optimal performances in a single run. This improves the overall performance of the beamformer. Attached Figure Description

[0087] Figure 1 Here is a block diagram of a multi-target beam pattern optimization algorithm in the spherical harmonic domain, as shown in the example.

[0088] Figure 2 The Pareto optimal surface corresponding to the optimal beam weights output by the optimization algorithm in this example;

[0089] Figure 3 The output of the optimized algorithm for this embodiment is a two-dimensional polar coordinate pattern of a beamformer. Detailed Implementation

[0090] The present invention will be further described below with reference to the embodiments and accompanying drawings, but this is not intended to limit the scope of the invention.

[0091] Example

[0092] An optimization method for multi-target beamformers in the spherical harmonic domain includes the following steps:

[0093] S1. The three performance indicators of the beamformer—white noise gain, directivity factor, and overall sidelobe level—are formalized and expressed as three objective functions of a multi-objective optimization problem.

[0094] S2. Treat the distortion-free constraint of the beamformer array on the desired direction as an equality constraint in a multi-objective optimization problem;

[0095] S3. Set constraint values ​​for white noise gain and directionality factor. Combining the characteristics of the positive definite quadratic form of the expression for white noise gain and directionality factor, determine the range of beamforming weights and use them as the search space of the variables to be optimized in the multi-objective optimization problem.

[0096] S4. Finally, the NSGA-II algorithm with constraints is used to solve the proposed multi-objective optimization problem, obtain the optimal beamforming weights, and realize the optimization of the optimal beamformer in the spherical harmonic domain.

[0097] The expression for the white noise gain mentioned in S1 is:

[0098]

[0099] In the above formula, d is the beamforming weight vector in the spherical harmonic domain, and d T This indicates the transpose of d;

[0100] Matrix A is:

[0101] A = vv T (twenty two)

[0102] Where v T This represents the transpose of vector v, where vector v is:

[0103] v = v(0) = [1, 3, ..., (2N+1)] T (twenty three)

[0104] Matrix B is:

[0105]

[0106] Where M is the number of microphones in the array, diag(.) means constructing a diagonal matrix from the vectors in parentheses, b n n = 0, ..., N represents the modal intensity related to the array configuration. The array configuration includes rigid spheres and open spheres, and the corresponding modal intensities are shown below:

[0107]

[0108] Where j n and These are the nth-order spherical Bessel function and the nth-order spherical Hankel function of the second kind, respectively, j' n and Let i and k represent the first derivatives of the corresponding functions, i be the imaginary unit, k be the wave number, and a be the radius of the sphere array.

[0109] The expression for the directional factor mentioned in S1 is:

[0110]

[0111] In the formula, matrix C is:

[0112]

[0113] The expression for the sum of the sidelobe levels in S1 is as follows:

[0114]

[0115] In the formula Ω SL Represents the sidelobe range, I represents the total number of discrete sampling points in the sidelobe region, and θ i ,φ i Let represent the elevation angle and azimuth angle at the i-th sampling point, respectively.

[0116] The expression for the distortionless response constraint in S2 is:

[0117] h(d)=d T v(0)=1 (29).

[0118] S3 describes determining the range of beamforming weights. First, a minimum threshold is set for both the white noise gain and the directivity factor. Without loss of generality, these minimum thresholds are denoted as ε. b and ε a Combining equations (5), (10), and (13), we can obtain the following equation:

[0119]

[0120]

[0121] Then, simplifying and splitting both sides of the above equation, we get the following equation:

[0122]

[0123]

[0124] Finally, the beamforming weight range constrained by white noise gain and directivity factor is obtained as follows:

[0125]

[0126] l = -h (35)

[0127] In the above formula, vectors l and h represent the lower and upper bounds of the obtained beamforming weights, respectively, and min(.) represents the minimum value operation, which is achieved by adjusting ε. a and ε b The value of can control the range of the beamforming weight vector d to be optimized, enabling the algorithm to converge to the optimal solution more quickly.

[0128] The multi-objective optimization problem described in S4 is the beam optimization problem in the spherical harmonic domain, which can be rewritten as the following multi-objective optimization problem:

[0129] min(-WNG,-DF,SLS)

[0130] subject to h(d=d) T v(0)=1

[0131] l≤d≤h (36).

[0132] Reference Figure 1 The solution to the multi-objective optimization problem described in S4 includes the following steps:

[0133] Step 1: Set parameters:

[0134] Set the range of the side lobe Ω SL The number of discrete sampling points I for the sidelobe level, the configuration of the spherical microphone array (including the number of microphones, array radius, and other parameters), the frequency point f for beamforming, and the beamforming order N. order White noise gain threshold ε b Directionality factor threshold ε a Parameters related to the NSGA-II algorithm: including population size N pop Maximum number of generations N max Optimize the number of variables v to simulate the crossover exponent η of binary crossover. c And the mutation exponent ηm used for polynomial mutation, and the crossover probability p m (A value between 0 and 1 controls the probability of an individual mutating);

[0135] This embodiment provides an example to demonstrate the effectiveness of the proposed algorithm, specifically setting the beamforming order N. order =4, optimized frequency f=1000HZ, the spherical array is configured as a rigid sphere with 32 microphones evenly distributed on its surface with a radius of 0.042m, and the sidelobe range (the range of the angle between the sidelobe and the desired direction of arrival) is Ω. l =[35°:2.4°:180°] represents uniform sampling with a stride of 2.4°, optimizing for a number of variables v=5, η c =20,η m =100, pm =1 / v, ε a =ε b =1, N pop =400, N max =2000;

[0136] The second step is to initialize the population:

[0137] Initialize the first generation population by randomly generating N within the range defined by l and h. pop The instance d corresponding to the beamforming weight vector d n (n = 1, 2, ..., N) pop (i.e., generating N) pop Replace d with a real number. n d in n (instances of n = 0, ..., N), and these form the first generation population, where each beam in the population forms a weight vector instance d. n Also known as an individual in the population, set the current population generation N. c =1;

[0138] If N pop =3 means that 3 instances are generated randomly;

[0139]

[0140] The third step is to calculate the modified objective function value:

[0141] For each individual d in the current population n Constraint handling techniques are applied to each objective function dimension i to compute the modified objective function value F. i (d n );

[0142] Step 4: Calculate the Pareto level and congestion distance:

[0143] Using F i (d n For each individual d in the current population n Perform a non-dominated sort and assign Pareto levels to them. Then, in each Pareto level set, use F... i (d n ) Calculate d for each individual n Crowded distance;

[0144] Step 5: Selection of paternal genes.

[0145] Select N using binary tournament selection method for the current population. popAmong the individuals, the fitness comparison criteria are as follows: individuals with smaller Pareto scores have higher fitness; when Pareto scores are the same, individuals with greater crowding distances have higher fitness; and finally, individuals with higher overall fitness are selected.

[0146] Step 6: Producing offspring individuals:

[0147] Each time, without replacement, two individuals are randomly selected from the parent generation as the parent and the target, respectively. Simulated binary crossover and polynomial mutation are then performed on these individuals to produce offspring. N pop The parent individuals produced a total of N. pop Individual offspring;

[0148] Step 7: Update the current population:

[0149] The set of offspring individuals generated in step six and the current population are combined to form a new population of size 2N. pop Given a set, perform the operation in step four on that set, and then select N based on fitness. pop Individuals replace the current population to form a new generation of population, for generations N. c Add one;

[0150] Step 8: Judgment

[0151] Determine the current algebra N c Is it greater than the maximum number of generations N? max If it is greater than , skip to step nine; otherwise, skip to step three.

[0152] Ninth step output:

[0153] Output the set of feasible individuals in the current population. Each individual in the set is a beamforming weight that satisfies Pareto optimality in all measures.

[0154] Reference Figure 2 The Pareto optimal surface, composed of beamforming weights, is output by the optimization algorithm in the embodiment. Each point coordinate represents the corresponding value of a beamformer in each objective function dimension. As can be seen from the figure, the points on the Pareto optimal surface are basically evenly distributed on a straight line. Therefore, in practice, the required beamformer can be flexibly selected from the algorithm output according to the requirements of various performance indicators of the beamformer.

[0155] Reference Figure 3 The embodiment optimizes the algorithm by selecting a beamforming weight from the output set and plots a two-dimensional polar coordinate pattern. It can be seen from the figure that when the angle between the direction of arrival of the interference signal and the direction of arrival of the desired signal is greater than 60°, the beamformer can suppress it by less than -14dB, and the beam pattern of the beamformer is axisymmetric.

[0156] The optimization method of this invention enables the beamformer to achieve Pareto optimal beamforming weights in terms of overall sidelobe level, directivity factor, and white noise gain, without the need for complex parameter adjustment.

Claims

1. A method for optimizing multi-target beamformers in the spherical harmonic domain, characterized in that, Includes the following steps: S1. The three performance indicators of the beamformer—white noise gain, directivity factor, and overall sidelobe level—are formalized and expressed as three objective functions of a multi-objective optimization problem. S2. Treat the distortion-free constraint of the beamformer array on the desired direction as an equality constraint in a multi-objective optimization problem; S3. Set constraint values ​​for white noise gain and directionality factor. Combining the characteristics of the positive definite quadratic form of the expression for white noise gain and directionality factor, determine the range of beamforming weights and use them as the search space of the variables to be optimized in the multi-objective optimization problem. S4. Finally, the NSGA-II algorithm with constraints is used to solve the proposed multi-objective optimization problem, obtain the optimal beamforming weights, and realize the optimization of the optimal beamformer in the spherical harmonic domain. The expression for the white noise gain mentioned in S1 is: ； In the above formula, Weighting vector for beamforming in the spherical harmonic domain Indicates to Transpose; matrix for: ； in Represents a vector Transpose, and vector for: ； matrix for: ； in The number of microphones in the array. Indicates will The vectors in the matrix are constructed as a diagonal matrix. , , representing the modal intensity related to the array configuration, which includes rigid spheres and open spheres, with the corresponding modal intensities shown below: ； in and These are the nth-order spherical Bessel function and the nth-order spherical Hankel function of the second kind, respectively. and Let represent the first derivatives of the corresponding functions. The imaginary unit, For wave number, The radius of the ball array; The expression for the directional factor mentioned in S1 is: ； In the formula matrix for: ； The expression for the side lobe level in S1 is as follows: ； In the formula Represents the range of the side lobe. This represents the total number of discrete sampling points performed on the sidelobe region. They represent the first Elevation and azimuth angles at each sampling point; The distortion-free constraint expression in S2 is: ； S3 describes determining the range of beamforming weights. First, a minimum threshold is set for both the white noise gain and the directivity factor. Without loss of generality, these minimum thresholds are denoted as... and Combining equations (1), (6), and (9), we can obtain the following equation: （10）； （11）； Then, simplifying and splitting both sides of the above equation, we get the following equation: （12）； （13）； Finally, the beamforming weight range constrained by white noise gain and directivity factor is obtained as follows: （14）； （15）； In the above formula, the vector and These represent the lower and upper bounds of the obtained beamforming weights, respectively. This indicates the minimum value operation, which is performed by adjusting... and The value of the weight vector to be optimized controls the beamforming weight vector. The range of values ​​allows the algorithm to converge to the optimal solution more quickly.

2. The method for optimizing multi-target beamformers in the spherical harmonic domain according to claim 1, characterized in that: The multi-objective optimization problem described in S4 is the beam optimization problem in the spherical harmonic domain, which can be rewritten as the following multi-objective optimization problem: （16）。 3. The method for optimizing multi-target beamformers in the spherical harmonic domain according to claim 2, characterized in that: Solving the multi-objective optimization problem described in S4 involves the following steps: Step 1: Set parameters: Set the range of the side lobes The number of points for discrete sampling of sidelobe levels Configuration of the spherical microphone array, frequency points of beam design Beamforming order White noise gain threshold Directional factor threshold Relevant parameters of the NSGA-II algorithm: including population size Maximum number of generations Optimize the number of variables Crossover index used to simulate binary crossover and the variation index used for polynomial variation Crossover probability ; The second step is to initialize the population: Initialize the first generation population, in and Randomly generated within the specified range Beamforming weight vector Corresponding instance These form the first generation population, where each beam in the population forms a weight vector instance. Also known as an individual in the population, setting the current population generation. ; The third step is to calculate the modified objective function value: For each individual in the current population Constraint handling techniques are applied to each objective function dimension i to compute the modified objective function value. ; Step 4: Calculate the Pareto level and congestion distance: use For each individual in the current population Perform a non-dominated sort and assign Pareto levels to them, then use the following methods in each Pareto level set: Calculate each individual Crowded distance; Step 5: Selection of paternal genes. Select using binary tournament selection on the current population Among the individuals, the fitness comparison criteria are as follows: individuals with smaller Pareto scores have higher fitness; when Pareto scores are the same, individuals with greater crowding distances have higher fitness; and finally, individuals with higher overall fitness are selected. Step 6: Producing offspring individuals: Each time, two individuals are randomly selected from the parent generation without replacement, serving as the parent and the target, respectively. Simulated binary crossover and polynomial mutation are then performed on these individuals to produce offspring. The total number of parent individuals produced Individual offspring; Step 7: Update the current population: The set of offspring individuals generated in step six and the current population are combined to form a new population of [number]. The set, the operation in step four is performed on the set, and then the fitness is selected. Individuals replace the current population to form a new generation of population, which affects the number of generations. Add one; Step 8: Judgment Determine the current algebra Is it greater than the maximum number of generations? If it is greater than , skip to step nine; otherwise, skip to step three. Ninth step output: Output the set of feasible individuals in the current population. Each individual in the set is a beamforming weight that satisfies Pareto optimality in all measures.

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