CN108983201B - Robust broadband beam former design method based on probability constraint - Google Patents

Robust broadband beam former design method based on probability constraint Download PDF

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CN108983201B
CN108983201B CN201810813396.XA CN201810813396A CN108983201B CN 108983201 B CN108983201 B CN 108983201B CN 201810813396 A CN201810813396 A CN 201810813396A CN 108983201 B CN108983201 B CN 108983201B
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鲍彧
陈华伟
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Nanjing University of Aeronautics and Astronautics
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Abstract

The invention discloses a design method of a robust broadband beam former based on probability constraint, and belongs to the technical field of array signal processing. The method comprises the following steps: firstly, the method converts the probability problem into a convex optimization problem through inequality deformation; secondly, calculating to obtain a steady parameter under the condition of assuming known microphone mismatch error prior information as uncertain maximum limits of each mismatch error expectation and variance; then, determining a probability selection range according to the characteristic that the function of the convex function changes along with the parameters; and finally, obtaining the weight of the steady far-field broadband beam former with good passband frequency invariance performance through proper probability parameter selection. Under the same error condition, the invention not only has good robustness, but also has better pass band consistency gain, and ensures that the signal can not generate obvious distortion.

Description

Design method of robust broadband beam former based on probability constraint
Technical Field
The invention relates to a design method of a robust broadband beam former based on probability constraint, and belongs to the technical field of array signal processing.
Background
In audio and voice signal processing, the design of a beam former adopting a microphone array is an important research direction, and a fixed weight beam former is widely applied because the fixed weight beam former does not depend on received data, the design is simple and easy to realize, and the real-time performance is good. In fact, however, the fixed weight beamforming design is affected by the mismatch error of the microphones in practical environments, which results in a great reduction in the performance of the fixed weight beamformer, not only causing interference or noise not to be effectively suppressed, but also causing even severe attenuation of the desired signal. Designing a wideband fixed weight beamformer with robust performance is therefore an important research direction today.
Existing robust design methods require a priori information assuming known mismatch errors. Common known mismatch error prior information includes the probability density function of the mismatch error, the maximum perturbation range of the mismatch error, and the expectation and variance information of the mismatch error. In practical situations, the probability density function of the mismatch error is often not easy to obtain, and thus a design method using the probability density function as the prior information cannot be designed for the practical situations. The beamformer designed robustly based on the maximum perturbation range of the mismatch error has good robustness, but the problem of over conservation exists often, so that the filtering performance of the beam response is reduced, and the situation of signal attenuation distortion occurs in the pass band. But the expectation of mismatch error and variance information are also typically easier to estimate statistically in practical situations. Meanwhile, compared with the maximum disturbance situation, the design based on the mismatch error expectation and the variance is more suitable for the application of the actual situation, and the problem of conservatism existing in the design method based on the disturbance range is solved to a certain extent. However, the existing design method based on mismatch error expectation and variance information still has the problem of over conservation, and the situation that the expected signal attenuation is true occurs.
Disclosure of Invention
In order to solve the technical problems in the background art, the invention provides a robust wideband beam former design method based on probability constraint, which has the advantages of good stability and robustness, and simultaneously can ensure no distortion of a desired signal.
The invention adopts the following technical scheme for solving the technical problems:
a robust broadband beamformer design method based on probability constraint includes the following steps:
(1) constructing a design method of a steady broadband beam former using probability condition to constrain a stop band level according to a minimum and maximum optimization theory;
(2) rewriting the target function with the statistical information in the step (1) into a determined value target function according to a triangle inequality and a Cauchy Schwarz inequality;
(3) deforming the stop band constraint expression of the probability condition in the step (1) through a triangle inequality, and converting the constraint condition into a determined value constraint condition which is expressed by a convex function and is related to the probability parameter by utilizing a Gaussian probability inequality in statistical optimization;
(4) recording data collected by a microphone under a silent environment so as to estimate the approximate ranges of expectation and variance of the gain phase error and the position error of the microphone, calculating a specific expression result of related parameters in an objective function and a constraint condition, and further obtaining an optimization problem of determining an expression and parameters according to the step (2) and the step (3);
(5) and determining the selection range of the probability parameters according to the monotonicity and the functional properties of the stop band constraint conditional expression, and solving by using a convex optimization tool box to obtain the weight vector of the broadband beam former.
Step (1) the design method for constructing the robust broadband beam former with probability constraint based on the minimum maximization theory comprises the following steps:
Figure BDA0001739740620000031
w in the above equation is the weight vector of the beamformer, g (θ)pF) and g (. theta.)sF) beamformer steering vectors in the pass-band and the stop-band, respectively, Δ g (θ)pF) and Δ g (θ)sAnd f) is the steering vector component of the pass band and stop band mismatch error perturbation, Pd(θpF) the expected response in the passband, ΓsbIs a set stop band quantity parameter for controlling the stop band level of the beam former, p is a set probability parameter, E {. is expected calculation, Pr {. is probability operation, and s.t. is constraint condition.
In step (2), the objective function is transformed into the objective function of the deterministic value problem by using a trigonometric inequality and a cauchy schwarz's inequality:
Figure BDA0001739740620000032
where ε is a robust performance parameter related to the amount of microphone mismatch error, | | - | represents the Euclidean norm.
In step (3), the constraint condition for converting the probability inequality in the constraint condition into a determined value:
i.e. if the constraint condition Pr { | w is to be madeT[g(θs,f)+Δg(θs,f)]|≤ΓsbP is satisfied, only
Figure BDA0001739740620000041
μ in the above formulamaxAnd
Figure BDA0001739740620000042
parameters related to the expectation and variance of mismatch error, IMIs an M multiplied by M dimensional unit matrix, Lambda (f) is an L multiplied by L dimensional matrix, and the (k, L) th element is Lambda(k,l)(f)＝cos[2π(k-l)f/fs]P is a probability parameter, ΓsbIs the stop band level parameter.
The specific process in step (4) is as follows:
the expected and variance limits of the microphone gain error, phase error and position error are estimated by using the intrinsic parameters of the microphone and the measured data, namely
|E[Δam]|≤μa (4)
Figure BDA0001739740620000043
|E[Δdm]|≤μd (6)
Figure BDA0001739740620000044
Figure BDA0001739740620000045
Figure BDA0001739740620000046
Wherein mua，
Figure BDA0001739740620000047
And mudIs a priori desired information of gain, phase and position errors,
Figure BDA0001739740620000048
Figure BDA0001739740620000049
and
Figure BDA00017397406200000410
is a priori variance information of gain, phase and position errors, and Var {. is a variance. The concrete expression results of the related parameters in the objective function and the constraint condition are respectively calculated as
Figure BDA00017397406200000411
Figure BDA0001739740620000051
Figure BDA0001739740620000052
Where M is the number of array elements, L is the length of the filter following the array elements, fmaxIs the maximum frequency of the frequency band, and c is the speed of sound.
Further obtaining the optimization problem of determining the expression and the parameter according to the step (2) and the step (3) as
Figure BDA0001739740620000053
The invention has the following beneficial effects:
the adjustable parameter p is introduced in the design process of the invention, and the advantages are as follows:
1) the robust performance and the passband beam response performance of the beam former can be adjusted in the design process by adjusting the parameter p, so that the invention can more conveniently design the robust broadband beam former meeting the working requirement under the same prior condition.
2) From the analysis it can be determined that the method of adjustment of the parameter p is simple and intuitive. A smaller value of p may result in better wideband beamformer passband beam response performance, while a larger value of p may result in better wideband beamformer robustness performance.
Drawings
Fig. 1 is a linear array far-field broadband beamformer model designed by the present invention.
Fig. 2(a) is a side view of the beam response of the worst performance optimization method in example 1.
Fig. 2(b) is a beam response side view of the conventional stop band constraining method in example 1.
Fig. 2(c) is a side view of the beam response of the method of the invention in example 1.
Figure 3 is a WNG comparison of the present invention with the worst performance optimization method and the conventional stopband constraint method under the determined probability parameters in example 1.
FIG. 4(a) is a graph comparing the worst performance optimization method and the conventional stopband constraint method PIF under different probability parameters in comparative example 1
FIG. 4(b) is a graph comparing the worst performance optimization method and the minimum WNG of the conventional stopband constraint method under different probability parameters in comparative example 1
FIG. 4(c) is a graph comparing the average WNG of the worst performance optimization method and the conventional stopband constraint method under different probability parameters in comparative example 1
Fig. 5(a) is a graph comparing the worst performance optimization method and the conventional stopband constraint method PIF under different stopband parameters in comparative example 2.
Fig. 5(b) is a graph comparing the minimum WNG of the present invention with the worst performance optimization method and the conventional stopband constraint method under different stopband parameters in comparative example 2.
Fig. 5(c) is a graph comparing the average WNG of the worst performance optimization method and the conventional stopband constraint method for the invention of comparative example 2 at different stopband parameters.
Detailed Description
The technical scheme of the invention is further explained in detail by combining the attached drawings:
the invention relates to a design method of a far-field broadband beam former suitable for any microphone type, which is not generalized, and is explained by taking a uniform linear array as an example.
The invention is that M same omnidirectional microphones are arranged in an array element with equal space as shown in figure 1The method is researched on the basis of uniform microphone linear arrays with the distance d. Without loss of generality, we define the linear array on the x-axis and the position of the m-th array element is rm＝[xm,0]T,m＝0,...M-1，xmThe position of the x-axis coordinate is that the incidence angle of the sound source to the array is plane wave and the included angle between the sound source and the coordinate axis is theta. Each microphone element is followed by a finite long impulse response (FIR) filter with a tap length L. The beam response of a signal of frequency f is expressed as
P(θ,f)＝wTg(θ,f) (14)
Wherein
Figure BDA0001739740620000071
Is the steering vector of the far-field array,
Figure BDA0001739740620000072
representing a kronecker operation, w is the beam response weight coefficient, and T is the transpose operation. At the same time, the user can select the required time,
h(θ,f)＝[h0(θ,f),h1(θ,f),…,hM-1(θ,f)]T (15)
Figure BDA0001739740620000073
wherein f issIs the sampling frequency. Without loss of generality, the origin of the coordinate system is assumed as the reference point, so the transfer function h of the m-th array elementm(θ, f), M is 0, …, M-1 far field linear array with ideal transfer function expressed as
hm(θ,f)＝exp{-j2πfxmcosθ/c} (17)
Where c represents the speed of sound in air.
In practice, the microphone often has mismatch errors. When there are microphone gain, phase and position errors, the transfer function of the mth microphone is rewritten as
Figure BDA0001739740620000074
Wherein Δ am，
Figure BDA0001739740620000075
And Δ dmRespectively, the gain error, phase error and position error at the mth microphone. Δ hm(θ, f) represents a transfer function that contains only mismatch error terms, M is 0, …, M-1.
Thus, the far field broadband beam response in the presence of mismatch error is expressed as
Figure BDA0001739740620000076
Figure BDA0001739740620000077
The actual steering vector representing the mismatch error in the presence of a disturbance component of the mismatch error is
Figure BDA0001739740620000081
In the formula
Δh(θ,f)＝[Δh0(θ,f),Δh1(θ,f),...,ΔhM-1(θ,f)] (21)
Vectors of transfer functions are formed for each array element of the mismatch error disturbance component.
An optimization problem can therefore be given for a design method with stop-band constraints assuming a robust design with known mismatch error expectation and variance part information as shown in (1). The inequality transformation of the cost function in (1) is as follows:
Figure BDA0001739740620000082
i.e., the result of equation (2). Wherein g (theta)pAnd f) denotes the beamformer steering vector in the passband, Pd(θpF) expected response in the passband, Δ g (θ)p,f)Is the steering vector portion of the passband mismatch error perturbation, of the formula
Figure BDA0001739740620000083
Hence robust parameters
Figure BDA0001739740620000084
Wherein M is the number of array elements of the array, L is the tap length of the filter connected behind the array elements, fmaxIs the maximum frequency of the operating band and c is the speed of sound.
The compound (10) can be obtained according to (23).
According to the gaussian probability inequality: assuming that a real random variable ξ obeys a unimodal distribution, the assumption without loss of generality is that its center is the mean μ and its standard deviation is σ, for a set parameter κ ≧ 0 of known size, there are
Figure BDA0001739740620000091
The stop band constraint in (1) can be modified to:
Figure BDA0001739740620000092
combining (24) and (25) to obtain (3).
The information required in the gaussian inequality is the mean and variance of the random variables. From (4) to (9)
Figure BDA0001739740620000093
Figure BDA0001739740620000094
According to (26) and (27), (11) and (12) can be obtained.
For the defined function (8), assuming that the weight vector w is fixed, it can be shown that the resulting function f (w, p) is a continuous monotonically increasing function of the variable p. That means for f (w, p). ltoreq.ΓsbIn this inequality, the smaller the set probability parameter p is, the larger the selectable range of the weight vector w satisfying the inequality is. This means that for the optimization problem (2), a smaller p has a larger range of possible solutions, which means that the objective function can achieve better results, i.e. the passband response of a smaller beamformer is closer to the desired value.
The performance metrics of the wideband beamformer were evaluated as follows:
white Noise Gain (WNG): used for evaluating the stability of the beam forming device, the larger the value is, the more stable the value is, the specific calculation formula is
Figure BDA0001739740620000101
Wherein: thetadAt a desired angle of incidence, wm,lThe weighting coefficient is the first filter tap after the m array, and l is the first tap of the FIR filter.
The smaller the value of the pass band frequency invariance index (PIF), the better the frequency invariance performance of the pass band of the beam former is, and the specific calculation formula is
Figure BDA0001739740620000102
Wherein: n is a radical ofpThe number of discrete points representing the angle of the pass band,
Figure BDA0001739740620000103
is the nth discrete pass band angle point value, K is the discrete point number of the design frequency band, f(k)For the k-th rational frequency point value,
Figure BDA0001739740620000104
for the nth angle dispersion, the kth frequency dispersion corresponds to the beam response value, and
Figure BDA0001739740620000105
the invention will be further illustrated with reference to some specific examples and to the scale.
Example 1
Pass band frequency non-deformation energy comparison graph and WNG comparison graph of robust design method based on probability constraint, robust design method based on conventional stop band constraint optimization and conventional worst performance optimization design method under different conditions
Consider a linear array consisting of 7 array elements, the array element spacing d is 0.04M, and the number of filter taps following each array element is L20. Setting the sampling frequency fs8000Hz and a frequency band range of [1500,3500 ] is considered]Hz, the angular range of the pass band is [80 degrees ], 100 degrees]The angle range of the stop band is [0 DEG, 60 DEG ]]∪[120°,180°]. Desired response of passband Pd(θp,f)＝exp{-j2πf(L-1)fs -1/2} and a stopband parameter Γsb-6 dB. For microphone error information, we assume | E [ Δ a ]m]|≤0.02，
Figure BDA0001739740620000111
|E[Δdm]|≤0.0007，|Var[Δam]|≤0.015，
Figure BDA0001739740620000112
And | Var [ Δ d [ ]m]|≤2×10-6m2。K＝40，N p20. The results of the beam response of its beamformer were obtained by 100 averaging experiments. First, based on the setting parameters, we calculate the values of (3), (4) and (5) to be epsilon 0.9778, mumax＝0.0826，
Figure BDA0001739740620000113
The parameter epsilon 0.9778 in the conventional stop band constraint optimization is consistent with the probability constraint method. The method is optimized for worst performance where parameter ε is 2.9323. For the probabilistic constraint method, settingp is 0.36. Fig. 2(a) is a side view of a beam pattern of the worst performance optimization method, fig. 2(b) is a side view of a beam pattern of the conventional stopband constraint optimization method, and fig. 2(c) is a side view of a beam pattern of the probability constraint method. It is clear from the performance indication in the figure that the probabilistic constraint method has the best passband frequency invariant performance. Further PIF for each process is given by table 1. The best performance of the constant pass band frequency can be more clearly shown according to the numerical values of the PIF in the table 1. While in figure 3 we show the performance display of WNG for these three methods. It can be seen from the figure that the worst performance optimization method has the highest robust performance as a whole, and the WNG variation interval performances of the other two methods are equivalent, and the minimum value is guaranteed to be more than 6dB, so that the worst performance optimization method has good robust performance.
TABLE 1
Method Worst performance optimization method Conventional stopband constraint method Probability constraint method
PIF -26.5437 -29.0179 -34.7200
Comparative example 1
And comparing the performances of the probability constraint method under different probability parameter selections with the performances of the worst performance optimization method and the conventional stop band constraint method.
The same as the initial parameters set in embodiment 1, in the probability constraint method, when the probability parameter p changes, each method PIF, the minimum WNG-to-probability variation graph and the average WNG-to-probability variation graph. The results of fig. 4(a) - (c) are the probability p increasing from 0.1 to 0.9 at 0.1 pitch, increasing by 0.999 to approximate the result of 1. The frequency invariable performance of the passband wave beam response of the broadband wave beam former designed by the probability constraint method is gradually reduced; as the probability increases, the overall trend of the minimum WNG and the average WNG increases, which indicates that the overall trend of the robust performance increases. This illustrates that the choice of probability parameter affects the beam response passband performance and the robust performance of its design. By properly selecting the probability, a result which meets the design requirements in two aspects can be found. On the other hand, compared with other two methods, the probability constraint method has the advantages that the passband frequency invariant performance is always superior to the worst performance optimization method, but the WNG of the probability constraint method is lower than that of the worst performance optimization method; compared with the conventional stop band constraint method, the probability constraint method can obtain better pass band performance by changing p within a certain range, and simultaneously ensures that the minimum WNG is higher than that of the conventional stop band constraint method; or higher WNG results than conventional stopband constraint methods while maintaining comparable passband performance. Thus, by appropriately adjusting the value of the probability parameter p, a beamformer with better passband performance can be obtained when p is smaller, and a beamformer with better robust performance can be obtained when p is larger.
Comparative example 2
And comparing the performances of the probability constraint method under different stop band parameter selections with the performances of the worst performance optimization method and the conventional stop band constraint method.
The same initial parameters as set in comparative example 1, when the probability parameters p in the probability constraint method are 0.1,0.5 and 0.9, respectively, and PIF, minimum WNG versus probability plot and average WNG versus stop band parameter plot for the worst performance optimization method and the conventional stop band constraint method. The results in FIGS. 5(a) - (c) are ΓsbResults of the performance change for each method were-2 dB, -4dB, -6dB, -8dB and-10 dB, respectively. It can be found from the results shown in the figure that the method proposed by the present invention is easier to obtain better passband performance with lower probabilityAn energy beamformer, while a higher WNG, i.e., a robust performance beamformer, can be obtained with a higher probability. On the other hand, by properly selecting and adjusting the probability parameter p, we can always find the beamformer design weights with better performance than the existing methods.
Comparative example 3
In practice, the beamformer designed in practical measurement example 1 processes the practical measurement signals and compares the results.
The actual measurement experiment is being carried out in an anechoic chamber having dimensions of 5.5 m.times.3.3 m.times.2.4 m. The microphone array used was the same as in example 1, and 7-element arrays were selected. The playing signal is a mixed sinusoidal signal with the frequency of 1500,2000,2500,3000 and 3500Hz, and the distance from the sound source to the array is 2.5m, so that the playing signal can be regarded as a far-field condition. And the NI USB-6363 acquisition card is used in the receiving end to acquire data at 8000 Hz. The final acquisition data is illustrated with data at 90 ° passband, 40 ° stopband. The microphone signal at the center of the array is assumed to be the reference signal. Table 2 shows the ratio of the spectral energy of the output signal to the input reference signal. As can be seen from the table, the method proposed by the present invention can obtain the best signal amplitude in the exemplary method in the expected incident direction while ensuring effective suppression of the signal from the stop band direction, and the signal has almost no attenuation, which indicates that the present invention can ensure good operation performance in practical use.
TABLE 2
Figure BDA0001739740620000141
Combining the results of the examples and comparative examples, the present invention provides greater flexibility in use and allows to obtain beamformers meeting different working requirements by adjusting the probability parameters. With appropriate probability parameter selection, better passband performance and good robust performance can be obtained compared to existing methods. Therefore, the method has certain practical value.
The above examples and comparative examples are only illustrative of the technical idea of the present invention, and are not intended to limit the scope of the present invention, and any modifications made on the basis of the technical solution according to the technical idea of the present invention are within the scope of the present invention.

Claims (5)

1. A robust wideband beamformer design method based on probability constraints, comprising the steps of:
(1) constructing a design method of a steady broadband beam former using probability condition to constrain a stop band level according to a minimum and maximum optimization theory;
(2) rewriting the target function with the statistical information in the step (1) into a determined value target function according to a triangle inequality and a Cauchy Schwarz inequality;
(3) deforming the stop band constraint expression of the probability condition in the step (1) through a triangle inequality, and converting the constraint condition into a determined value constraint condition which is expressed by a convex function and is related to the probability parameter by utilizing a Gaussian probability inequality in statistical optimization;
(4) recording data collected by a microphone under a silent environment so as to estimate the approximate ranges of expectation and variance of the gain phase error and the position error of the microphone, calculating a specific expression result of related parameters in an objective function and a constraint condition, and further obtaining an optimization problem of determining an expression and parameters according to the step (2) and the step (3);
(5) and determining the selection range of the probability parameters according to the monotonicity and the functional properties of the stop band constraint conditional expression, and solving by using a convex optimization tool box to obtain the weight vector of the broadband beam former.
2. A method for robust wideband beamformer design based on probability constraints as claimed in claim 1 wherein: in the step (1), a design method of a robust broadband beam former with probability constraint is constructed based on a minimum maximization theory, and the method specifically comprises the following steps:
Figure FDA0003473473890000011
s.t.Pr{|wT[g(θs,f)+Δg(θs,f)]|≤Γsb}≥p
w in the above equation is the weight vector of the beamformer, g (θ)pF) and g (. theta.)sF) beamformer steering vectors in the pass-band and the stop-band, respectively, Δ g (θ)pF) and Δ g (θ)sAnd f) is the steering vector component of the pass band and stop band mismatch error perturbation, Pd(θpF) the expected response in the passband, ΓsbIs a set stop band quantity parameter for controlling the stop band level of the beam former, p is a set probability parameter, E {. is expected calculation, Pr {. is probability operation, and s.t. is constraint condition.
3. A robust wideband beamformer design method based on probability constraints according to claim 1 wherein: in step (2), the objective function is transformed into the objective function of the deterministic value problem by using the trigonometric inequality and the cauchy schwarz inequality:
Figure FDA0003473473890000012
where w is the weight vector of the beamformer, ε is the robust performance parameter related to the amount of microphone mismatch error, g (θ)pAnd f) denotes the beamformer steering vector in the passband, Pd(θpAnd f) is the expected response within the passband, | | · | |, represents the euclidean norm.
4. A robust wideband beamformer design method based on probability constraints according to claim 1 wherein: in step (3), the probability inequalities in the constraints are converted into the constraints of determined values:
i.e. if the constraint condition Pr { | w is to be madeT[g(θs,f)+Δg(θs,f)]|≤ΓsbP is satisfied, only
Figure FDA0003473473890000021
W in the above equation is the weight vector of the beamformer, μmaxAnd
Figure FDA0003473473890000022
parameters related to the expectation and variance of mismatch error, IMIs an M × M dimensional unit matrix, Λ (f) is an L × L dimensional matrix, and the (k, L) th element is Λ(k,l)(f)＝cos[2π(k-l)f/fs]，fsFor the sampling frequency, p is a probability parameter, ΓsbIs a set stop band number parameter, g (theta)sF) beamformer steering vectors in the stopband, Δ g (θ)sAnd f) is the steering vector portion of the stopband mismatch error perturbation.
5. A robust wideband beamformer design method based on probability constraints according to claim 1 wherein: the specific process in step (4) is as follows:
from data collected by the microphone in a silent environment, approximate ranges of expectations and variances of microphone gain phase error and position error are estimated, i.e.
|E[Δam]|≤μa
Figure FDA0003473473890000023
|E[Δdm]|≤μd
Figure FDA0003473473890000024
Figure FDA0003473473890000025
Figure FDA0003473473890000026
Wherein Δ am，
Figure FDA0003473473890000027
And Δ dmRespectively representing a gain error, a phase error and a position error on the mth microphone; mu.sa，
Figure FDA00034734738900000213
And mudIs a priori desired information of gain, phase and position errors,
Figure FDA0003473473890000028
and
Figure FDA0003473473890000029
is the prior variance information of gain, phase and position errors, and Var {. cndot } represents the variance;
and then calculating concrete expression results of related parameters in the objective function and the constraint condition as follows:
Figure FDA00034734738900000210
Figure FDA00034734738900000211
Figure FDA00034734738900000212
where M is the number of array elements, L is the length of the filter following the array elements, fmaxIs the maximum frequency of the frequency band, c is the speed of sound;
and obtaining the optimization problem of determining the expression and the parameter according to the step (2) and the step (3) as
Figure FDA0003473473890000031
Figure FDA0003473473890000032
Wherein: w is the weight vector of the beamformer, IMIs an M multiplied by M dimensional unit matrix, Lambda (f) is an L multiplied by L dimensional matrix, and the (k, L) th element is Lambda(k,l)(f)＝cos[2π(k-l)f/fs]，fsFor the sampling frequency, p is a probability parameter, ΓsbIs a set stop band number parameter, [ epsilon ] is a robust performance parameter related to the amount of microphone mismatch error, [ mu ]maxAnd
Figure FDA0003473473890000033
is a parameter, g (theta), related to the expectation and variance of mismatch errorpF) denotes the beamformer steering vector in the pass band, g (θ)sF) beamformer steering vectors in the stopband, Δ g (θ)sAnd f) is the steering vector component of the stopband mismatch error perturbation, Pd(θpAnd f) is the expected response within the passband, s.t. represents the constraint.
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