A method for evaluating micro-crack damage using nonlinear lamb waves

Abstract

The application discloses a method for evaluating micro crack damage by using nonlinear Lamb waves and belongs to the technical field of ultrasonic nondestructive testing. x The method comprises the following steps: obtaining signals based on a sensor array to form a signal library, obtaining signals in the signal library and performing band-pass filtering on the obtained signals to form a signal matrix, solving a correlation matrix R x based on the signal matrix, obtaining a corresponding eigenvector E N based on the correlation matrix and the eigenvector, obtaining a spatial intensity spectrum, positioning and imaging micro cracks through the spatial intensity spectrum to construct an image library, obtaining a spatial intensity angle, and constraining a maximum angle as a beamforming pointing angle θ0, performing spatial filtering by using a beamforming algorithm, and completing positioning of imaging in the image library. The application can quickly image and position and evaluate micro cracks.

Description

Technical Field

[0001] This invention belongs to the field of ultrasonic nondestructive testing technology, specifically relating to a method for evaluating microcrack damage using nonlinear Lamb waves. Background Technology

[0002] Plate-like structures are widely used in production and construction in fields such as machinery, chemical engineering, construction, and aerospace. Under long-term stress and loads, they are prone to fatigue damage, which can then develop into macroscopic damage that impairs structural strength, leading to structural failure and production accidents. Therefore, early microcrack detection of structures is an important means of ensuring safe production.

[0003] Lamb waves are guided waves that propagate in thin plates, characterized by long propagation distances and sensitivity to internal defects, and are widely used in the field of non-destructive testing of plate structures. Traditional Lamb wave testing techniques are mostly based on linear theory, detecting damage by observing changes in the sound velocity, attenuation, phase, and reflection coefficient of the echo caused by defects. However, they are not sensitive to early subwavelength microcracks. Microcracks in structures exhibit contact nonlinearity effects, generating high-order harmonics in transmitted and reflected waves. Summary of the Invention

[0004] Purpose of the invention: To provide a method for evaluating microcrack damage using nonlinear Lamb waves, which solves the above-mentioned problems existing in the prior art.

[0005] Technical solution: A method for evaluating microcrack damage using nonlinear Lamb waves, comprising the following steps:

[0006] A signal library is formed by acquiring signals based on a sensor array;

[0007] Acquire signals from the signal library and perform band filtering on the acquired signals to form a signal matrix. Solve for the correlation matrix R based on the signal matrix. x and obtain R x The corresponding eigenvector E N Spatial intensity spectrum is obtained based on correlation matrix and eigenvector, and image library is constructed by localizing microcracks through spatial intensity spectrum imaging;

[0008] The spatial intensity angle is obtained and the maximum angle is constrained to the beamforming pointing angle θ0. Spatial filtering is performed using the beamforming algorithm to complete the localization of the image in the image database.

[0009] Preferably, the Lamb wave dispersion curve is obtained, and a predetermined mode is selected from the signal library as the excitation signal and the excitation signal is modulated; based on the wavelength of the Lamb excitation signal, a predetermined array spacing is selected, and a sensor array is arranged on the surface of the aluminum plate according to the array spacing. The Lamb wave excitation signal belonging to S1 is symmetrically applied at the excitation sensor to excite the S0 mode Lamb wave in the aluminum plate. The response signal of the sensor array is received and stored as a signal matrix X(t).

[0010] Preferably, based on the Lamb wave dispersion curve, within a range where the second harmonic is less than the cutoff frequency of other modes, the Lamb wave S0 mode signal with low dispersion and high speed is selected as the excitation.

[0011] Preferably, a rectangular window and a Hanning window are used to modulate the Lamb wave excitation signal. The modulation method is as follows:

[0012]

[0013] In the formula: ε(t) is the step function; f c The center frequency of the excitation signal; t represents the phase of the signal; N represents the number of peaks of the modulated sine wave; f represents the frequency of the Lamb wave; and t represents the time variable.

[0014] Preferably, the spatial intensity spectral function P(θ) is calculated as follows:

[0015]

[0016] In the formula: A(θ) represents the array direction matrix, E N H represents the noise eigenvector, and H represents the matrix conjugate transpose.

[0017] The preferred multi-input signal algorithm (MUSIC) is:

[0018] Suppose a linear array consists of N sub-elements spaced d apart, and M incoherent signal sources in the far field, with the angle θ between each source and the normal direction of the first element of the array being the angle of incidence. The signal matrix X(t) received by the array is expressed as:

[0019] X(t) = A(θ)S(t) + N(t)

[0020] In the formula: the expressions for A(θ), S(t) and N(t) are as follows:

[0021]

[0022] S(t)=[S1(t)S2(t)…S M (t)] T

[0023] N(t) = [n1(t) n2(t) … n N (t)]

[0024] Where: A(θ) is the direction matrix of the linear array, the signal matrix emitted by M far - field incident sources is S(t), and the angles formed with the normal direction of the first element of the array are Θ = [θ1 θ2 … θ M T , N(t) is the noise matrix;

[0025] The cross - correlation matrix of the received signals is R x = E[XX H . Here, H represents the conjugate transpose of the matrix. Assuming that the noise is uncorrelated, zero - mean, additive white Gaussian noise, the cross - correlation matrix is expressed as follows:

[0026] R x = E[(A(θ)S(t)+N(t))(A(θ)S(t)+N(t)) H

[0027] = AE[SS H A H + E[NN H

[0028] = AR[[ID=第38]] S A H + R N

[0029] Where, R S = E[SS H represents the signal correlation matrix, R N = σ 2 I represents the noise correlation matrix, σ 2 is the variance of the noise signal, and I is the M×M identity matrix. Now assume that the eigenvalues of the covariance matrix R x are {λ1, λ2, …, λ N}, and the corresponding eigenvectors are {q1, q2, …, q N} in turn. In the case where the multiple sources are uncorrelated and the number of sources M < N, AR S A H is positive semi - definite, and there are M larger eigenvalues and N - M smaller eigenvalues with a value of σ 2 . Sort the eigenvalues of the matrix R x in descending order, that is, λ1 > λ2 > … > λ M > … λ N ​​​, where M larger eigenvalues correspond to the signal space, and the remaining N - M smaller eigenvalues correspond to the noise space. The eigenvectors corresponding to the signal eigenvalues and the noise eigenvalues are the signal eigenvector and the noise eigenvector, respectively;

[0030] According to the definitions of eigenvalues and eigenvectors:

[0031] R x q i = λ i q i

[0032] For M < k < N, there is

[0033] R x q k = λ k q k = σ 2 q k

[0034] Multiply both the left and right sides of the cross-correlation matrix R x by q k It can be

[0035] R x q k = (AR S A H + σ 2 I)q k

[0036] = AR s A H q k + σ 2 q k

[0037] After simplification, we get AR s A H q k = 0. Multiply both the left and right sides of it by as follows:

[0038]

[0039] That is:

[0040] A H q k = 0

[0041] The above equation shows that the noise eigenvector and the array direction selection vector are orthogonal; by using this orthogonality property, by solving the noise eigenvector E N corresponding to the cross-correlation matrix of the array signal, searching for the azimuth angle of arrival of the signal source, generating the spatial spectrum, and realizing the direction estimation of the signal arrival.

[0042] Preferably, the beamforming algorithm is used for spatial filtering, and the calculation method is as follows:

[0043]

[0044] In the formula, n represents the array index, N represents the number of array elements, and w n Here, θ0 represents the pointing angle of the formed beam, f represents the frequency of the Lamb wave, and c represents the wave velocity of the Lamb wave. n (t) represents the received signal of the corresponding array element, e represents the natural constant, d represents the array element spacing, and j represents the imaginary unit.

[0045] Preferably, the nonlinear parameter β′ of the array signal after beamforming algorithm processing is calculated as follows:

[0046]

[0047] In the formula, A1 and A2 are the fundamental frequency intensity and secondary frequency intensity of the signal, respectively.

[0048] Beneficial Effects: This invention relates to a method for assessing microcrack damage using nonlinear Lamb waves. Based on a sensor array, it achieves large-scale, baseline-free, and precise localization imaging of microcracks, solving the problem of microcrack localization imaging in aluminum plates. This enables structural health detection of early damage in structures, ensuring production and construction safety. Simultaneously, combined with beamforming algorithms, it assesses the degree of damage to microcracks at different locations. Damage localization can be completed solely by acquiring signals from the sensor array. It exhibits good resistance even in noisy environments. Applied to practical detection, it is beneficial for predicting and assessing the health of structures, providing accurate information for structural health evaluation. Attached Figure Description

[0049] Figure 1 This is a flowchart illustrating the steps of the damage localization method of the present invention;

[0050] Figure 2 This is a simulation model diagram of the present invention;

[0051] Figure 3 The excitation signal used in this invention and its corresponding spectrum diagram are shown.

[0052] Figure 4 This is a time-domain diagram of the received signals of each element in array A of the present invention;

[0053] Figure 5 The spectrum diagrams of the array of the present invention are shown for the plate when it is undamaged and when it contains microcracks.

[0054] Figure 6 This is a comparison of the time and frequency domains of the array signal before and after filtering according to the present invention;

[0055] Figure 7 This is the spatial spectrum obtained after processing by the MUSIC algorithm of this invention;

[0056] Figure 8 Images showing the localization and imaging effects of microcracks at different locations;

[0057] Figure 9 Here is a flowchart of the beamforming algorithm processing;

[0058] Figure 10 The spectrum and nonlinear parameter diagrams of microcrack beamforming with different widths and thicknesses are shown. Detailed Implementation

[0059] In practical applications, the applicant discovered that Lamb waves, guided waves propagating in thin plates, possess the characteristics of long propagation distance and sensitivity to internal defects, and are widely used in the field of nondestructive testing of plate structures. Traditional Lamb wave detection techniques are mostly based on linear theory, detecting damage by observing changes in the sound velocity, attenuation, phase, and reflection coefficient of the echo caused by defects. However, they are insensitive to early subwavelength microcracks. Microcracks in structures exhibit contact nonlinearity effects, generating high-order harmonics in transmitted and reflected waves. To address these issues, a method for evaluating microcrack damage using nonlinear Lamb waves was invented, which effectively solves the aforementioned problems.

[0060] like Figures 1 to 5 As shown, a method for evaluating microcrack damage using nonlinear Lamb waves is proposed, which uses a sensor array to acquire signals to form a signal library.

[0061] Acquire signals from the signal library and perform band filtering on the acquired signals to form a signal matrix. Solve for the correlation matrix R based on the signal matrix. x and obtain R x The corresponding eigenvector E N Spatial intensity spectrum is obtained based on correlation matrix and eigenvector, and image library is constructed by localizing microcracks through spatial intensity spectrum imaging;

[0062] The spatial intensity angle is obtained and the maximum angle is constrained to the beamforming pointing angle θ0. Spatial filtering is performed using the beamforming algorithm to complete the localization of the image in the image database.

[0063] In a further embodiment, the Lamb wave dispersion curve is obtained, and a predetermined mode is selected from the signal library as the excitation signal and modulated; for example... Figure 2As shown, two linear arrays are arranged on a plate-like material. Based on the wavelength of the Lamb excitation signal, a predetermined array spacing is selected. A sensor array is then arranged on the aluminum plate surface according to the array spacing. For the sensor array, the element spacing d must satisfy... This ensures that there are no sidelobes during beamforming and guarantees good focusing capability. The Lamb wave excitation signal belonging to mode S1 is symmetrically applied at the excitation sensor to generate a Lamb wave of mode S0 in the aluminum plate. The response signal of the sensor array is received and stored as a signal matrix X(t). In this embodiment, a Hanning window is used for modulation. For single-frequency excitation, the excitation signal is expressed as follows:

[0064]

[0065] Where ε(t) is the step function; f c The center frequency of the excitation signal; t represents the phase of the signal; N represents the number of peaks of the modulated sine wave; f represents the frequency of the Lamb wave; and t represents the time variable.

[0066] In a further embodiment, based on the Lamb wave dispersion curve, within the cutoff frequency range where the second harmonic is less than that of other modes, the Lamb wave S0 mode signal with low dispersion and high speed is selected as the excitation.

[0067] In a further embodiment, the linear array is assumed to consist of N sub-element elements spaced d apart, and there are M incoherent signal sources in the far field, with the angle θ between these sources and the normal direction of the first element of the array being the angle of incidence; the signal matrix X(t) received by the array is expressed as:

[0068] X(t) = A(θ)S(t) + N(t)

[0069] The expressions for A(θ), S(t), and N(t) are as follows:

[0070]

[0071] S(t)=[S1(t)S2(t)…S M (t)] T

[0072] N(t)=[n1(t)n2(t)…n N (t)]

[0073] Where A(θ) is the direction matrix of the linear array, and the signal matrix S(t) emitted by the M far-field incident sources forms an angle Θ = [θ1 θ2 … θ] with the normal direction of the first element of the array. M ] T N(t) is the noise matrix.

[0074] The cross-correlation matrix of the received signals is R x = E[XX H , where H represents the conjugate transpose of the matrix; assuming that the noise is uncorrelated, zero-mean, additive white Gaussian noise, the cross-correlation matrix is expressed as follows:

[0075] R x = E[(A(θ)S(t) + N(t))(A(θ)S(t) + N(t)) H

[0076] = AE[SS H A H + E[NN H

[0077] = AR S A H + R N

[0078] In the formula, R S = E[SS H represents the signal correlation matrix, R N = σ 2 I represents the noise correlation matrix, σ 2 is the variance of the noise signal, and I is the M×M identity matrix. Now assume that the eigenvalues of the covariance matrix R x are {λ1, λ2,..., λ N}, and the corresponding eigenvectors are {q1, q2,..., q N} in sequence. In the case where the multiple signal sources are uncorrelated and the number of signal sources M < N, AR S A H is positive semi-definite, and there are M larger eigenvalues and N - M smaller eigenvalues with the value of σ 2 . Sort the eigenvalues of the matrix R x in descending order, that is, λ1 > λ2 >... > λ M >... λ N , where M larger eigenvalues correspond to the signal space, and the remaining N - M smaller eigenvalues correspond to the noise space. The eigenvectors corresponding to the signal eigenvalues and the noise eigenvalues are the signal eigenvectors and the noise eigenvectors respectively;

[0079] According to the definition of eigenvalues and eigenvectors

[0080] R x q i = λ i q i

[0081] For M < k < N, there is

[0082] ​​R x q k =λ k q k =σ 2 q k

[0083] Let the cross-correlation matrix be R. x Multiply by q on both sides k Can

[0084] R x q k =(AR) S A H +σ 2 I)q k

[0085] =AR s A H q k +σ 2 q k

[0086] Simplify to get AR s A H q k =0, multiply by both sides. as follows:

[0087]

[0088] Right now:

[0089] A H q k =0

[0090] The above equation shows that the noise eigenvector and the array orientation selection vector are orthogonal. Utilizing this orthogonality, the noise eigenvector E corresponding to the cross-correlation matrix of the array signals can be solved. N The azimuth of arrival of the signal source is searched to generate a spatial spectrum, thereby achieving direction-of-arrival estimation based on the spatial intensity spectrum function formula:

[0091]

[0092] After obtaining the spatial spectrum of the microcrack location from the two arrays, cross-location is performed, and the intersection point is the location of the microcrack, which enables rapid location of the microcrack.

[0093] After locating the microcrack, in a further embodiment, a beamforming algorithm based on a time-delay weighted summation method is used to achieve spatial filtering of the microcrack signal, and beamforming of the array signal is achieved by the following formula;

[0094]

[0095] In the formula, n represents the array index, N represents the number of array elements, wn is the pointing weighting coefficient, representing the pointing angle of the formed beam, f represents the frequency of the Lamb wave, c represents the wave velocity of the Lamb wave, and x represents the beam velocity. n (t) represents the received signal of the corresponding array element, e represents the natural constant, d represents the array element spacing, and j represents the imaginary unit.

[0096] Experimental results

[0097] The experiment selected an aluminum plate as the sample to be tested. Specific material performance parameters are shown in Table 1, where E represents Young's modulus, υ represents Poisson's ratio, and ρ represents density. The modeling is as follows: Figure 2 As shown, the model dimensions are 600mm × 600mm × 4mm; the defect type is buried microcrack damage. The geometric center of the plate is taken as the origin, with the length and width directions corresponding to the x and y axes of the coordinate system, respectively, and the thickness direction as the z-axis. Two sensor arrays, A and B, each consisting of 7 elements, are established with the first array element at point A1 (-50, 50) and point B1 (-50, -50), and an element spacing of 5mm. A symmetrical load is applied at coordinate (-50, 0), exciting a single S0 mode Lamb wave in the plate. The excitation load signal is a sinusoidal signal with a center frequency of 150kHz modulated by a 5-cycle Hanning window. The time-domain waveform and frequency-domain spectrum are shown below. Figure 3 As shown, Figure 3(a) is the time-domain signal diagram, and (b) is the spectrum diagram. In this paper, the group velocity of the fundamental frequency signal is approximately 5440 m·s⁻¹, and the wavelength is 36.3 mm. The wave velocity of the 300 kHz nonlinear Lamb wave second harmonic signal generated by the interaction with the microcrack is approximately 5210 m·s⁻¹, and the wavelength is 17.4 mm.

[0098] Table 1 Performance parameters of aluminum materials

[0099]

[0100] Figure 4 The waveforms represent the time-domain waveforms of the signals received by array A in the microcrack model. Different curves represent the signals received by different array elements in the array, which are respectively divided into excitation direct wave packet, microcrack echo wave packet, and boundary reflection wave packet. Figure 5 The spectrum of the echo signal received by the first array element is shown. It can be seen that, compared to the lossless plate, the microcracked plate generates a nonlinear signal at the second harmonic caused by the microcrack. The received echo signal is a waveform containing both fundamental and second harmonic components. To locate the microcrack, the fundamental signal, which cannot characterize the microcrack, needs to be filtered out. A Butterworth bandpass filter was designed and used to extract the second harmonic signal. The time-domain waveforms and spectrum diagrams of the echo signal before and after passing through the filter are shown in Figure 6. (a) is the time-domain comparison diagram, and (b) is the frequency-domain comparison diagram. Figure 6 This demonstrates that the bandpass filter effectively filters out the fundamental signal while preserving the second harmonic component relatively well.

[0101] After processing the filtered microcrack echo signal using the MUSIC algorithm, the angle between the microcrack location and the normal direction of the array can be obtained. To achieve damage localization imaging, two sensor arrays at different locations are needed to obtain different orientation angles of the microcrack relative to the two arrays, and then the microcrack location is achieved using a cross-localization method. The spatial spectra of the signals received by the two linear arrays A and B, with A1 and B1 as their first elements, are calculated respectively, as shown below. Figure 7 As shown, the two curves represent the actual incident angles of the microcracks relative to arrays A and B, respectively.

[0102] Using the spatial spectra extracted from arrays at different locations, the MUSIC algorithm was used to locate and image microcracks at (72.5, 2.5) and (22.5, -2.5), respectively. The results are as follows: Figure 8 As shown in (a)-(b), the '+' sign indicates the location of the actual damage. This figure demonstrates that the present invention can obtain excellent microcrack localization imaging by using the MUSIC algorithm to process nonlinear Lamb wave signals.

[0103] The microcrack located at coordinates (72.5, 2.5) is analyzed. First, finite element analysis is performed on the microcrack model mentioned above, and spatial filtering of the array signal is performed using a beamforming algorithm. Figure 9 The waveforms are obtained after filtering and beamforming the fundamental and second harmonic signals in the original signal received by the array. Simulation analysis was performed for different microcrack heights (h) and thicknesses (δ). The first set of simulations kept the microcrack length l and thickness δ constant (h = 5.0 mm, δ = 300 nm), selecting crack models with different heights (h = 1.5 mm, 2.0 mm, 2.5 mm, 3.0 mm, 3.5 mm). The second set of simulations kept the microcrack length l and height h constant (l = 5.0 mm, h = 3.0 mm), selecting crack models with different thicknesses (δ = 100 nm, 150 nm, 200 nm, 300 nm, 400 nm). The nonlinear parameters after beamforming are shown below. Figure 10 As shown, (a) and (b) are the spectrum diagrams and nonlinear parameter diagrams of the signals after processing microcracks of different heights, and (c) and (d) are the spectrum diagrams and nonlinear parameter diagrams of the signals after processing microcracks of different thicknesses. It can be seen that the beamforming process of this invention provides a good microcrack damage assessment capability.

[0104] The preferred embodiments of the present invention have been described in detail above. However, the present invention is not limited to the specific details in the above embodiments. Within the scope of the technical concept of the present invention, various equivalent transformations can be made to the technical solutions of the present invention, and these equivalent transformations all fall within the protection scope of the present invention.

Claims

1. A method for evaluating microcrack damage using nonlinear Lamb waves, characterized in that, It includes the following steps: A signal library is formed based on signals acquired by a sensor array. The Lamb wave dispersion curve is obtained, and a predetermined mode is selected as the excitation signal from the signal library and modulated. Based on the wavelength of the Lamb excitation signal, a predetermined array spacing is selected, and the sensor array is arranged on the surface of an aluminum plate according to the array spacing. The Lamb wave excitation signal belonging to mode S1 is symmetrically applied at the excitation sensor, exciting the Lamb wave of mode S0 in the aluminum plate. The response signals of the sensor array are received and stored as a signal matrix. ; Acquire signals from the signal library and perform band filtering on the acquired signals to form a signal matrix. Solve for the correlation matrix based on the signal matrix. and obtain Corresponding feature vectors Spatial intensity spectrum is obtained based on correlation matrix and eigenvector, and image library is constructed by localizing microcracks through spatial intensity spectrum imaging; Obtain the spatial intensity angle and constrain the maximum angle to the beamforming pointing angle. The beamforming algorithm is used for spatial filtering to locate the images in the image database. Among them, the beamforming algorithm performs spatial filtering and the calculation method is as follows: ； In the formula, n represents the array index, N represents the number of array elements, and w n To indicate the weighting coefficients, This indicates the pointing angle of the formed beam, f represents the frequency of the Lamb wave, and c represents the wave velocity of the Lamb wave. For the received signal of the corresponding array element, e represents the natural constant, d represents the array element spacing, and j represents the imaginary unit; Nonlinear parameters of array signals after beamforming algorithm processing The calculation method is as follows: ； In the formula, , These represent the fundamental frequency intensity and the second frequency intensity of the signal.

2. The method for evaluating microcrack damage using nonlinear Lamb waves according to claim 1, characterized in that, According to the Lamb wave dispersion curve, within the range where the second harmonic frequency is less than the cut-off frequencies of other modes, select the Lamb wave S0 mode signal with low dispersion and high velocity as the excitation.

3. The method for evaluating microcrack damage using nonlinear Lamb waves according to claim 1, characterized in that, Use rectangular window and Hanning window to modulate the Lamb wave excitation signal, and the modulation method is as follows: ； In the formula: f is a step function; c The center frequency of the excitation signal; t represents the phase of the signal; N represents the number of peaks of the modulated sine wave; f represents the frequency of the Lamb wave; and t represents the time variable.

4. The method for evaluating microcrack damage using nonlinear Lamb waves according to claim 1, characterized in that, Spatial intensity spectrum function The calculation method is as follows: ； In the formula: Represents the array direction matrix. H represents the noise feature vector, and H represents the matrix conjugate transpose.

5. The method for evaluating microcrack damage using nonlinear Lamb waves according to claim 4, characterized in that, The Multiple Signal Classification (MUSIC) algorithm is as follows: Suppose a linear array consists of N sub-elements spaced d apart, and M incoherent signal sources in the far field. The angle between these sources and the normal direction of the first element of the array is given. The incident angle is the signal matrix received by the array. The expression is as follows: ； In the formula: , and The expressions are as follows: ； ； ； In the formula: The direction matrix is ​​given by the linear array, and the signal matrix emitted by the M sources incident from the far field is given by... And each of them forms an angle with the normal direction of the first element of the array. , Noise matrix; The cross-correlation matrix of the received signal is Where H denotes the conjugate transpose of the matrix, and assuming the noise is uncorrelated zero-mean additive Gaussian white noise, the cross-correlation matrix is ​​expressed as follows: ； Wherein, represents the signal correlation matrix, represents the noise correlation matrix, is the variance of the noise signal, is the M×M identity matrix. Now assume that the covariance matrix has eigenvalues of , and the corresponding eigenvectors are successively . In the case where multiple signal sources are uncorrelated and the number of signal sources M < N, is positive semi-definite, and has M larger eigenvalues and N - M smaller eigenvalues with a value of . Sort the eigenvalues of the matrix in descending order, that is , where M larger eigenvalues correspond to the signal space, and the remaining N - M smaller eigenvalues correspond to the noise space. The eigenvectors corresponding to the signal eigenvalues and the noise eigenvalues are the signal eigenvectors and the noise eigenvectors respectively; According to the definition of eigenvalues and eigenvectors: ； For M < k < N, there is: ； The cross-correlation matrix is Riding on both sides Can: ； Simplify to get To ride together on the left and right ,as follows: ； That is: ； The above equation shows that the noise eigenvector and the array orientation selection vector are orthogonal; utilizing this orthogonality, the noise eigenvector corresponding to the cross-correlation matrix of the array signals can be solved. The system searches for the azimuth angle of the signal source, generates a spatial spectrum, and estimates the direction of signal arrival.

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