A magnitude-directed filtering method

Abstract

The application discloses a magnitude-directed filtering method, and belongs to the technical field of mechanical fault diagnosis and modal analysis. The method comprises the following steps: acquiring an original signal and performing filtering processing on the original signal; constructing a Hankel matrix based on the filtered signal; based on the equivalent relationship between SVD and PCA, constructing a magnitude filter to perform decomposition processing on the Hankel matrix; determining the amplitude order position of a target frequency component in the amplitude spectrum of the original signal according to the amplitude spectrum of the original signal; selecting a group of components corresponding to the target frequency component from a plurality of frequency components according to the order position to perform signal reconstruction, and obtaining a reconstruction matrix; and recovering a time-domain signal from the reconstruction matrix, that is, a filtered signal of the target frequency component. According to the order of the characteristic frequency in the amplitude spectrum, the specified frequency is selected for filtering, and only the order of the characteristic frequency in the amplitude spectrum needs to be known, so that the specified characteristic frequency can be accurately extracted or eliminated.

Description

Technical Field

[0001] This invention relates to the field of mechanical fault diagnosis and modal analysis technology, and more particularly to an amplitude-guided filtering method, and more specifically to an amplitude-guided filtering method based on the equivalence principle of singular value decomposition and principal component analysis. Background Technology

[0002] The vibration signals of rotating machinery contain crucial information reflecting the working state of its components. By first collecting the vibration signals of the rotating machinery components and then analyzing them using signal processing methods, characteristic factors reflecting the operating state of the components can be extracted. Using these characteristic factors, combined with existing fault diagnosis methods, various fault modes can be accurately diagnosed.

[0003] Singular Value Decomposition (SVD) has wide applications in the field of vibration signal processing. By constructing a one-dimensional signal into a Hankel matrix, it can effectively reduce noise in nonlinear signals. A key challenge in SVD is selecting the order of the effective singular values. Several solutions exist to address this problem, such as unsupervised dynamic clustering algorithms, algorithms that filter out fractal dimensions of the signal, and singular value accumulation methods.

[0004] Principal Component Analysis (PCA) is a method for extracting the principal components of a dataset. Its purpose is to remove correlations between input random vectors and highlight implicit characteristics in the original data. Due to its ability to reduce dimensionality or compress data, it is now widely used in image processing, neural networks, fault diagnosis, and other fields.

[0005] While many existing methods have achieved remarkable results in certain specific situations, they lack broad applicability. Many methods achieve the goal of separating signals into different subspaces by setting a threshold for singular values ​​using a specific algorithm, but fail to reveal the correspondence between signal frequency components, singular values ​​in singular value decomposition, and eigenvalues ​​in principal component analysis in terms of quantity, amplitude, and arrangement order.

[0006] For example, the related technology, CN108021871A, entitled "A Feature Frequency Extraction Method Based on Principal Component Analysis," analyzed its characteristics and studied the correspondence between eigenvalues ​​and signal frequency components in the PCA algorithm in terms of quantity and amplitude. Through experimental testing and theoretical derivation, it was proven that each effective frequency component in the signal generates at most two non-zero eigenvalues, which are closely arranged one after the other; the number of effective eigenvalues ​​is related to the number of frequencies in the signal, but independent of the signal's amplitude, frequency, and phase; and the larger the amplitude of a frequency component, the larger the two effective eigenvalues ​​it generates. However, this study did not prove the relationship between singular values ​​in singular value decomposition and eigenvalues ​​in principal component analysis, did not study the correspondence between singular values ​​in singular value decomposition and signal frequency components in terms of quantity and amplitude, did not summarize the relationship between singular values / eigenvalues ​​and signal length, and the analysis of the feature frequency extraction algorithm was also incomplete.

[0007] Therefore, this invention proposes an amplitude-guided filtering method. Summary of the Invention

[0008] This invention addresses the shortcomings of existing technologies by providing an amplitude-guided filtering method.

[0009] The technical solution of the present invention to solve the above-mentioned technical problems is as follows: An amplitude-guided filtering method includes the following steps: Acquire the original signal and perform filtering on it; The filtered signal is constructed in Hankel matrix form to obtain the Hankel matrix X; Based on the equivalence relationship between singular value decomposition and principal component analysis, an amplitude filter is constructed to decompose the Hankel matrix X; wherein, the equivalence relationship is predefined as: the eigenvalues ​​of the covariance matrix of the Hankel matrix X. Singular values ​​of the Hankel matrix X satisfy , n represents the column number of the Hankel matrix X, and the left singular vector obtained during singular value decomposition is equivalent to the eigenvector obtained during principal component analysis; the original signal is decomposed into multiple frequency components by an amplitude filter; Based on the amplitude spectrum of the original signal, determine the amplitude sorting position k of the target frequency component in the amplitude spectrum of the original signal; Based on the amplitude sorting position k, a set of components corresponding to the target frequency component is selected from multiple frequency components for signal reconstruction to obtain a reconstruction matrix; The time-domain signal is recovered from the reconstructed matrix, and the time-domain signal is the filtered signal of the target frequency component extracted from the original signal.

[0010] Furthermore, the construction of the amplitude filter decomposes the Hankel matrix X, specifically including: Based on the aforementioned equivalence relation, singular value decomposition or principal component analysis is selected to decompose the Hankel matrix X; When using singular value decomposition, perform singular value decomposition on the Hankel matrix X to find its singular values. Arranged in descending order as follows , ,…, Construct the singular value distribution spectrum and obtain the left and right singular matrices; based on the singular values... The singular value distribution spectrum is sorted, and two singular values ​​corresponding to the target frequency component and their corresponding left and right singular vectors are selected for reconstruction. When using principal component analysis, find the covariance matrix C of the Hankel matrix X and its eigenvalues. Arranged in descending order as follows , ,…, Construct the eigenvalue distribution plot of the covariance matrix and obtain the eigenvectors corresponding to each eigenvalue; based on the eigenvalues... In the sorting of the eigenvalue distribution map of the covariance matrix, two eigenvalues ​​corresponding to the target frequency component and their corresponding eigenvectors are selected for reconstruction.

[0011] Further, based on the sorting position k, a set of components corresponding to the target frequency component is selected for reconstruction, including: If the amplitude of a certain frequency component in the amplitude spectrum of the original signal is sorted as k, then the (2k-1)th and 2kth singular values / eigenvalues ​​and their corresponding left and right singular vectors / eigenvectors in the singular value distribution spectrum / covariance matrix eigenvalue distribution map are selected for signal reconstruction.

[0012] Furthermore, the equivalence relation is constructed through the following steps: For the Hankel matrix X, calculate its covariance matrix. And through the covariance matrix Perform eigenvalue decomposition to obtain eigenvalues ;in: The superscript T indicates transpose; X express m × n A matrix of order 1, m Let X be the row number of the Hankel matrix. Simultaneously, singular value decomposition is performed on the Hankel matrix X to obtain singular values. : ,in, U It is a left singular matrix; V It is a right singular matrix; S It is a diagonal matrix; Based on the above relationship, the component results obtained by singular value decomposition or principal component analysis can be mapped to each other, that is, the eigenvalues ​​and singular values ​​obtained by the same Hankel matrix X under the two decompositions are consistent; the amplitude filter can select either singular value decomposition or principal component analysis for signal decomposition and reconstruction.

[0013] Furthermore, the Hankel matrix X is constructed as follows: ; In the formula, X It is an m×n matrix. m Let X be the row number of the Hankel matrix. The first element in the Hankel matrix X i Okay, number j The elements of the column are 1≤i≤m, 1≤j≤n; L is the length of the signal data, and L=m+n-1; when L is even, m=L / 2, n=L / 2+1; when L is odd, m=(L+1) / 2, n=(L+1) / 2.

[0014] Further, recovering the time-domain signal from the reconstructed matrix includes: The average value of each diagonal element of the reconstructed matrix is ​​calculated using the averaging method to recover the one-dimensional time-domain signal; or, the inverse transformation of the reconstructed matrix is ​​performed using the pseudo-inverse matrix method to recover the one-dimensional time-domain signal.

[0015] Furthermore, in the multiple frequency components obtained by the amplitude filter decomposition, each frequency component generates two non-zero singular values / effective eigenvalues, and the two non-zero singular values / effective eigenvalues ​​are arranged one after the other in the singular value distribution spectrum / covariance matrix eigenvalue distribution diagram. The number of non-zero singular values / effective eigenvalues ​​is only related to the number of frequency components and is independent of the amplitude, frequency, and phase of the signal. The order of non-zero singular values / effective eigenvalues ​​in the singular value distribution spectrum / covariance matrix eigenvalue distribution map is determined by the amplitude of the signal frequency components. The larger the amplitude of the frequency component, the larger the non-zero singular values / effective eigenvalues ​​it produces, and the earlier it is arranged.

[0016] Furthermore, for signals of different lengths, the singular values ​​are related to the data length and the amplitude of the characteristic frequencies as follows: ; in, and Let L be the singular value corresponding to a certain frequency component, and L be the signal length. The amplitude corresponding to a certain frequency component; Based on the equivalence relationship between singular values ​​and eigenvalues, the above equation can be transformed as follows: ; in, and It is the eigenvalue corresponding to a certain frequency component.

[0017] Furthermore, it also includes: When the target frequency component is submerged in noise in the amplitude spectrum of the original signal and its amplitude ranking position k cannot be determined, an auxiliary signal is injected into the original signal. The frequency of the auxiliary signal is the same as that of the target frequency component, so as to improve the amplitude ranking position of the target frequency component in the amplitude spectrum. In the new amplitude spectrum, the improved sorting position k' is determined, and the corresponding decomposition result is selected according to the improved sorting position k' for reconstruction, so as to obtain a reconstructed signal containing the target frequency component and the auxiliary signal; The auxiliary signal is subtracted from the reconstructed signal to obtain the filtered signal of the target frequency component. In summary, compared with the prior art, the beneficial effects of the above technical solution are: (1) The method of the present invention is to select a specified frequency for filtering based on the order of the characteristic frequencies in the amplitude spectrum. As long as the order of the characteristic frequencies in the amplitude spectrum is known, the specified characteristic frequencies can be accurately extracted or eliminated. Even if two frequency components differ by only 1 Hz, they can be successfully separated. This is difficult to achieve with frequency-oriented filters. (2) The method of the present invention is a zero-phase-shift frequency extraction method. The frequency components of the filtered signal almost completely overlap with those of the original signal, with no phase shift. This characteristic makes it different from filters that use convolution operations to achieve signal separation. If the filter does not have linear phase characteristics, the separated signal will exhibit phase distortion. Attached Figure Description

[0018] Figure 1 The singular value distribution spectra of each group of signals are constructed; where (a) is the singular value distribution spectrum containing one frequency component; (b) is the singular value distribution spectrum containing two frequency components; and (c) is the singular value distribution spectrum containing three frequency components. Figure 2 The following are time-domain and frequency-domain plots of the constructed simulation signal; where (a) is the time-domain waveform of the simulation signal; and (b) is the amplitude spectrum of the simulation signal. Figure 3 This is the singular value distribution spectrum obtained after processing the simulated signal using the amplitude filter algorithm; Figure 4 The time-frequency domain diagrams of three different signal components obtained by reconstructing the signal using each set of singular values ​​are shown below. (a) is the time-frequency domain diagram obtained by reconstructing the signal using the first and third sets of singular values; (b) is the time-frequency domain diagram obtained by reconstructing the signal using the second set of singular values; and (c) is the time-frequency domain diagram obtained by reconstructing the signal using the fourth set of singular values. Figure 5 This is a schematic diagram of the installation of eddy current displacement sensors at the journal positions of the rotor near the motor end and far from the motor end of the test bench. Figure 6 These are time-domain and frequency-domain plots of signals acquired near the motor end and far from the motor end; where (a) is the time-domain and frequency-domain plot acquired near the motor end; and (b) is the time-domain and frequency-domain plot acquired near the motor end. Figure 7 These are the feature value distribution diagrams of the signals acquired near the motor end and far from the motor end; where (a) is the feature value distribution diagram of the signals acquired near the motor end; and (b) is the feature value distribution diagram of the signals acquired far from the motor end. Figure 8 These are frequency diagrams after refining the characteristic frequencies of the signals acquired near the motor end and far from the motor end; where (a) is the frequency diagram after reconstructing the two sets of characteristic signals acquired near the motor end; and (b) is the frequency diagram after reconstructing the two sets of characteristic signals acquired far from the motor end. Figure 9 The images show the shaft center trajectory refined using amplitude filter algorithm, harmonic wavelet, and wavelet packet algorithms. (a) shows the shaft center trajectory synthesized from the vibration signal near the motor end after being filtered by the amplitude filter algorithm; (b) shows the shaft center trajectory synthesized from the vibration signal far from the motor end after being filtered by the amplitude filter algorithm; (c) shows the shaft center trajectory synthesized from the characteristic frequencies of the far-motor end data refined using harmonic wavelet; and (d) shows the shaft center trajectory obtained by refining the far-motor end data after performing an 8-level decomposition using Daubechies wavelet packet (db42). Figure 10 This is the spectrum after filtering out 50Hz using D11; Figure 11 These are the time-domain and frequency-domain plots and singular value distribution plots before and after adding frequency components when the feature frequency to be extracted is submerged; where (a) is the time-domain signal plot when the feature frequency to be extracted is submerged by noise after increasing noise; (b) is the amplitude spectrum plot when the feature frequency to be extracted is submerged by noise after increasing noise; (c) is the singular value distribution plot of the signals corresponding to (a) and (b); (d) is the time-domain signal plot after injecting an auxiliary signal into the original signal; (e) is the amplitude spectrum plot after injecting an auxiliary signal into the original signal; and (f) is the singular value distribution plot obtained by processing the signal after injecting the auxiliary signal using an amplitude filter algorithm. Figure 12 The two diagrams are the frequency spectrum of the signal to be extracted separated from the noise and the axis trajectory synthesized using the signal; wherein, (a) is the frequency spectrum of the signal to be extracted separated from the noise; and (b) is the axis trajectory synthesized using the separated signal to be extracted. Detailed Implementation

[0019] The principles and features of the present invention are described below with reference to all the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.

[0020] This invention discloses an amplitude-guided filtering method.

[0021] Example 1 Reference Figures 1-12 An amplitude-guided filtering method includes the following steps: Acquire the original signal and perform filtering on it; The filtered signal is constructed in Hankel matrix form to obtain the Hankel matrix X; Based on the equivalence between Singular Value Decomposition (SVD) and Principal Component Analysis (PCA), an amplitude filter is constructed to decompose the Hankel matrix X. The equivalence relationship is predefined as the eigenvalues ​​of the covariance matrix of the Hankel matrix X. Singular values ​​of the Hankel matrix X satisfy , n represents the column number of the Hankel matrix X, and the left singular vector obtained during SVD is equivalent to the eigenvector obtained during PCA; the original signal is decomposed into multiple frequency components by an amplitude filter; Based on the amplitude spectrum of the original signal, determine the amplitude sorting position k of the target frequency component in the amplitude spectrum of the original signal; Based on the sorting position k, select a set of components corresponding to the target frequency component from the multiple frequency components obtained by decomposition to reconstruct the signal and obtain the reconstruction matrix. The time-domain signal is recovered from the reconstructed matrix, and the time-domain signal is the filtered signal of the target frequency component extracted from the original signal.

[0022] The method of this invention selects a specified frequency for filtering based on the order of the characteristic frequencies in the amplitude spectrum. Knowing only the order of the characteristic frequencies in the amplitude spectrum is sufficient to accurately extract or eliminate the specified characteristic frequencies. Even if two frequency components differ by only 1 Hz, they can be successfully separated, which is difficult to achieve with frequency-oriented filters.

[0023] For a fixed frequency power frequency interference, the two eigenvalues / singular values ​​corresponding to that frequency in the eigenvalue distribution map / singular value distribution spectrum of the covariance matrix, as well as the corresponding characteristic frequency / left singular matrix and right singular matrix, are selected for reconstruction. Then, the reconstructed characteristic signal is subtracted from the original signal to filter out the interference of the power frequency signal.

[0024] The above content will be explained in detail below: S1: Acquire the raw signal The original signal is then filtered, that is, the interference components in the original signal are filtered out and the effective signal components are retained. Then, the Hankel matrix X is constructed from the signal after the interference components are removed.

[0025] Specifically, the expression for the Hankel matrix X is as follows: ; In the above formula, X for m × n A matrix of order 1; for X The first in the vector i Okay, number j The elements of the column, 1 ≤ i ≤m, 1≤ j ≤n; m is the number of rows in the matrix, n is the number of columns in the matrix; L is the length of the signal data, L=m+n-1.

[0026] S2: Based on the equivalence between Singular Value Decomposition (SVD) and Principal Component Analysis (PCA), an amplitude filter is constructed to decompose the Hankel matrix X.

[0027] Specifically, the equivalence relation is predefined as: the eigenvalues ​​of the covariance matrix of the Hankel matrix X. Singular values ​​of the Hankel matrix X satisfy , n represents the column number of the Hankel matrix X, and the left singular vector obtained during SVD is equivalent to the eigenvector obtained during PCA; the original signal is decomposed into multiple frequency components by an amplitude filter.

[0028] When using Singular Value Decomposition (SVD), the Hankel matrix X is subjected to SVD to obtain its singular values. Arranged in descending order as follows , ,…, Construct the singular value distribution spectrum and obtain the left singular matrix. U And right singular matrix V According to singular values The singular value distribution spectrum is sorted, and two singular values ​​corresponding to the target frequency component and their corresponding left and right singular vectors are selected for reconstruction.

[0029] ; In the formula, the superscript T indicates transpose; U It is a left singular matrix. ,make ,in, ; V It is a right singular matrix. ,make , ;and, U and V All are orthogonal matrices; S is a diagonal matrix. Its expression is as follows: ; In the formula, , , ,…, Let X be the singular values ​​of the Hankel matrix X, and ≥……≥ .

[0030] Based on the order of singular values ​​in the singular value distribution spectrum, two singular values ​​corresponding to a specific frequency and their corresponding left and right singular vectors are selected for reconstruction. If the amplitude of a certain frequency in the original signal amplitude spectrum is ordered as k, then the left and right singular vectors corresponding to the (2k-1)th and 2kth singular values ​​in the singular value distribution spectrum are selected for signal reconstruction, resulting in a new matrix. .

[0031] A magnitude filter based on principal component analysis can also be constructed. Compared with the magnitude filter based on SVD, it only requires constructing the Hankel matrix X of the filtered signal, finding the covariance matrix C of the Hankel matrix X, and then finding its eigenvalues. Arranged in descending order as follows: , ,…, Find the eigenvectors corresponding to the eigenvalues. Then based on the eigenvalues The distribution of the matrix is ​​reconstructed by selecting two eigenvalues ​​and their corresponding eigenvectors corresponding to a specific frequency, following the same pattern, to obtain a new matrix. .

[0032] S3: Reconstruct the matrix The mean of the original signal is superimposed to obtain a matrix. Then, the averaging method is used to extract from the matrix Signal recovered .

[0033] Specifically, signals The expression is as follows: ; In the formula, To approximate the reconstructed matrix The Middle i line, number j Column elements; m Let n be the number of rows in the Hankel matrix X, and n be the number of columns in the Hankel matrix X. i,j,m,n,k All are positive integers.

[0034] For an amplitude filter based on singular value decomposition, the expression for singular value decomposition is as follows:

[0035] By utilizing the properties of orthogonal matrices, Equation 1 can be transformed into:

[0036] According to Equation 2, a component reconstructed in singular value decomposition can be expressed as follows:

[0037] In the formula, the left singular value vector That is eigenvectors; It is a right singular vector; Let X be the singular values ​​of the Hankel matrix X, and ≥……≥ According to singular values In the singular value distribution spectrum, the two singular values ​​corresponding to a specific frequency and their corresponding left and right singular vectors are selected for reconstruction. If the amplitude of a certain frequency in the original signal amplitude spectrum is sorted as k, then the left and right singular vectors corresponding to the (2k-1)th and 2kth singular values ​​in the singular value distribution spectrum are selected for signal reconstruction, resulting in a new matrix. .

[0038] For a magnitude filter based on principal component analysis, the Hankel matrix X is represented as... Principal component analysis of the Hankel matrix X yields the following results: l A new variable indicator , i=1,2…l ( l≤m The expression is as follows:

[0039] In the above formula, ; Let X be the row vector of the Hankel matrix X; the Hankel matrix X is... m × n Matrix; T is the transpose notation; Let C be the covariance matrix of the Hankel matrix X. i The eigenvectors corresponding to the eigenvalues ​​from largest to smallest; For feature vectors The first j One element, ; i , j All are positive integers.

[0040] According to the principle of principal component analysis, the characteristic equation of the covariance matrix C is as follows:

[0041] In the above formula, C is the covariance matrix; These are the eigenvalues ​​of the covariance matrix C; Is with The corresponding eigenvectors are arranged with the eigenvalues ​​in descending order. Based on eigenvalues In the eigenvalue distribution plot of the covariance matrix, two eigenvalues ​​and their corresponding eigenvectors corresponding to a specific frequency are selected for reconstruction. If the amplitude of a certain frequency in the original signal amplitude spectrum is sorted as k, then the eigenvectors corresponding to the (2k-1)th and 2kth eigenvalues ​​in the eigenvalue distribution plot of the covariance matrix are selected. Signal reconstruction is performed to obtain a new matrix. .

[0042] The reconstructed matrix The mean of the original signal is superimposed to obtain a matrix. That is, the matrix is ​​approximately reconstructed, and then the signal is recovered according to the matrix construction method.

[0043] In the Hankel matrix construction mode, a pseudo-inverse matrix is ​​used to recover the signal. The expression is ,in yes M The false rebellion, M It consists of m n-order identity matrices. Taking a matrix with m=3 and n=3 as an example, its expression is: ; Since the number of non-zero values ​​in M ​​is much smaller than the number of zeros, it is a sparse matrix. M For a sparse matrix, when m andn When the value of is large, if the pseudo-inverse matrix method is used to recover the signal, then the pseudo-inverse matrix is ​​calculated. The memory and time required will increase exponentially, therefore the averaging method is used here to reconstruct the matrix from an approximate value. Recovery signal The expression is as follows:

[0044] In the formula, To approximate the reconstructed matrix The Middle i line, number j Column elements; m Let n be the number of rows in the Hankel matrix X, and n be the number of columns in the Hankel matrix X. i,j,m,n,k All are positive integers, and the signal can be recovered according to Formula 6. .

[0045] This amplitude filter is an amplitude-oriented filter that selects a specified frequency for filtering based on the order of the characteristic frequencies in the amplitude spectrum. As long as the order of the characteristic frequencies in the amplitude spectrum is known, the specified characteristic frequency can be accurately extracted or eliminated. Even if two frequency components differ by only 1 Hz, they can be successfully separated.

[0046] For a fixed frequency power frequency interference, the two eigenvalues / singular values ​​corresponding to that frequency in the eigenvalue distribution map / singular value distribution spectrum of the covariance matrix, as well as the corresponding characteristic frequency / left singular matrix and right singular matrix, are selected for reconstruction. Then, the reconstructed characteristic signal is subtracted from the original signal to filter out the interference of the power frequency signal.

[0047] Example 2 To explore the intrinsic relationship between the number of non-zero singular values ​​and the number of frequencies of a signal, signals with different amplitudes, frequencies, and phases are constructed, as shown in the following equation:

[0048] In the above formula, k is the number of effective frequency components; The amplitude of the signal; The frequency of the signal; The phase of the signal is represented; 4096 data points were collected.

[0049] (1) First, we will explore the case when k=1. We will construct three sets of signals, each of which contains only one frequency component. Select 10Hz, 40Hz, and 70Hz; The distribution of singular values ​​when the phase is 10, 40, or 70 is as follows: Figure 1 As shown in (a) in the example, this shows the distribution of the first 50 singular values.

[0050] from Figure 1 As can be seen from (a) in the diagram, when k=1, each constructed signal contains only one effective frequency component, and the amplitude... ,frequency and phase They are all different. From the singular value distribution spectra of the three sets of signals, it can be seen that each signal produces two non-zero singular values. and Furthermore, these two singular values ​​are closely arranged one after the other. Also, the two non-zero singular values ​​generated by these three sets of signals are of equal magnitude.

[0051] (2) Secondly, let's explore k=2, For the cases of 1 and 0.8, three sets of signals are constructed similarly, each containing two effective frequency components. Take 10 and 20, 40 and 50, 70 and 80, phase Taking values ​​of 10 and 20, 40 and 50, and 70 and 80, the resulting singular value distributions are as follows: Figure 1 As shown in (b) in the image; from Figure 1 As can be seen from (b) in the figure, when k=2, When the values ​​are 1 and 0.8, each of the three constructed signal groups contains two effective frequency components. The amplitudes of the common frequency components in the three signal groups are... Same size, but corresponding frequency and phase They are different. As can be seen from the singular value distribution spectra of the three sets of signals, each signal generates two sets of non-zero singular values, and each set of singular values ​​contains two singular values. and They are arranged closely one after the other. It can also be seen that, although the signals are different, the two sets of non-zero singular values ​​generated by the three sets of signals are sequentially equal.

[0052] Furthermore, by comparison, it can be found that the magnitudes of the first group of non-zero singular values ​​are the same in both images. The signal at k=2 is the signal at k=1 with the addition of a frequency component with an amplitude of 0.8. Therefore, it can be determined that the first group of non-zero singular values ​​in the images... and It is determined by amplitude The first group of non-zero singular values ​​is generated by the frequency component that is 1, while the second group of non-zero singular values ​​is generated by the amplitude. It was generated at a frequency component of 0.8.

[0053] (3) Subsequently, we will discuss k=3. Considering the values ​​of 1, 0.8, and 0.6, construct three sets of signals, each containing three effective frequency components. The frequency of the first set of signals... The frequencies are 10Hz, 20Hz, and 30Hz; the frequencies of the second group of signals. The frequencies are 40Hz, 50Hz, and 60Hz; the frequencies of the third group of signals The frequencies are 70Hz, 80Hz, and 90Hz. The phase of the first group of signals... The phases are 10, 20, and 30; the phases of the second group of signals. The phases are 40, 50, and 60; the phases of the third group of signals. The constructed signals and the resulting singular value distributions for 70, 80, and 90 are as follows: Figure 1 As shown in (c); from Figure 1 As can be seen from (c) in the figure, when k=3, When the values ​​are 1, 0.8, and 0.6, each of the three constructed signal sets contains three effective frequency components. From the singular value distribution spectra of the three signal sets, it can be seen that each signal set produces three sets of non-zero singular values, and each set of singular values ​​contains two singular values. and They are arranged closely one after the other; and the three sets of singular values ​​produced by each signal are the same size as the singular values ​​produced by the other two signals.

[0054] Comparing the attached figures, it can be seen that the magnitudes of the first group of non-zero singular values ​​in these three figures are exactly the same. and It is determined by amplitude This is generated by the frequency component of 1. Comparing the cases when k=2 and k=3, we can see that the magnitudes of the second group of non-zero singular values ​​in both graphs are exactly the same. As mentioned earlier, the second group of non-zero singular values ​​when k=2... and It is generated by a frequency component with an amplitude of 0.8, hence the second set of non-zero singular values ​​when k=3. and It is also generated by a frequency component with an amplitude of 0.8; the third set of singular values ​​at k=3. and This is generated by a frequency component with an amplitude of 0.6. The same result can be obtained by further increasing the effective frequency components of the signal.

[0055] Therefore, the following conclusions can be drawn: (1) Each frequency component in the signal generates two adjacent non-zero singular values; (2) The number of non-zero singular values ​​of a signal depends only on the number of frequency components and not on the signal amplitude. ,frequency and phase Irrelevant; (3) The order of the non-zero singular values ​​of the signal in the singular value distribution spectrum is determined by the amplitude of the signal frequency. The larger the amplitude of a certain frequency, the larger the two non-zero singular values ​​it produces, and the earlier they appear in the order of the frequencies.

[0056] Based on the relationship between frequency and singular values, a magnitude filter method based on SVD is proposed, with the following steps: (1) For a given signal First, the DC component of the original signal is filtered out by FFT transformation, and then the Hankel matrix X is constructed using the signal with the DC component removed. (2) Perform singular value decomposition on the Hankel matrix X and find its singular values. Arranged in descending order And obtain the left singular matrix. And right singular matrix ; (3) Based on singular values In the singular value distribution spectrum, select two singular values ​​corresponding to a specific frequency and their corresponding left and right singular vectors for reconstruction. If the amplitude of a certain frequency in the original signal amplitude spectrum is sorted as k, then select the left and right singular matrices corresponding to the (2k-1)th and 2kth singular values ​​in the singular value distribution spectrum for signal reconstruction to obtain a new matrix. (4) Superimpose the reconstructed matrix with the mean of the original matrix to obtain the matrix. ; (5) From the matrix The signal is recovered using the averaging method. This signal is the characteristic frequency component that needs to be extracted.

[0057] If the eigenvalues ​​in principal component analysis are analyzed using the above method, similar conclusions can be obtained, and a feature frequency extraction method based on PCA can be proposed, namely the content disclosed in patent CN108021871A "A Feature Frequency Extraction Method Based on Principal Component Analysis".

[0058] Example 3 This study explores the relationship between singular values ​​in singular value decomposition and eigenvalues ​​in principal component analysis, given m random vectors. x 1, x 2… x m Each vector x i Contains n samples, that is x i =( x i1 , x i2 … x inIf the signal is such that an m×n order Hankel matrix can be constructed, then the signal can be used to construct a Hankel matrix. X As shown in the following formula: ; On the one hand, the Hankel matrix X Represented as and the Hankel matrix X Principal component analysis yields the following results: A new variable indicator , The expression is as follows:

[0059] In Equation 8, It can be known that Hankel matrix X The covariance matrix in descending order of size The eigenvectors corresponding to each eigenvalue.

[0060] As can be seen from the principle of principal component analysis, the covariance matrix The characteristic equation is as follows:

[0061] In Equation 9, For matrix eigenvalues, For eigenvalues The corresponding feature vector. It is a symmetric positive definite matrix, and its eigenvectors are mutually orthogonal, i.e. If signal processing is required, signal reconstruction is performed according to Formula 8 by multiplying both sides of the equation on the left. And summing them up, we get:

[0062] Based on cumulative contribution rate It can achieve data dimensionality reduction, if based on cumulative contribution rate Before choosing By reconstructing the eigenvectors, an approximate matrix can be obtained:

[0063] Then a component of the reconstructed PCA signal can be expressed as:

[0064] On the other hand, performing singular value decomposition on the Hankel matrix X yields:

[0065] In Equation 13, It is a left singular matrix. It is a right singular matrix. and They are all orthogonal matrices. S It is a diagonal matrix. .make , ,in , Then equation 13 can be transformed into:

[0066] Then a component reconstructed by SVD can be represented as:

[0067] Multiply both sides of equation 15 by the left side and will Substituting into equation 14, we get:

[0068] It is evident that Equations 12 and 16 are very similar in form. This is because when performing singular value decomposition on the Hankel matrix X, the left singular value vector obtained is... That is The eigenvectors, i.e. The covariance matrix of the Hankel matrix X It can be represented as:

[0069] Pair Perform the following transformation:

[0070] In Equation 18 We can obtain:

[0071] Substituting equation 17 into the above equation, we get:

[0072] From Equation 9 and Substituting into equation 20, we get:

[0073] Therefore, the eigenvalues ​​can be obtained. With singular values The following relationship exists:

[0074] That is, there is equivalence between the singular values ​​in singular value decomposition and the eigenvalues ​​in principal component analysis.

[0075] Example 4 To explore the relationship between singular values / eigenvalues ​​and signal frequency, we first investigate the quantitative relationship between non-zero singular values / effective eigenvalues ​​and signal frequency components.

[0076] For a given signal containing one frequency component , , , These represent the signal's amplitude, frequency, and phase, respectively. (Based on the sampling frequency...) Discretized signals ,structure OK The column's Hankel matrix is ​​as follows:

[0077] According to Euler's formula, equation 23 can be transformed into the following equation:

[0078] Rewrite equation 24 as a sum of two equations, namely:

[0079] From the above equation, we can see that the rank of the first term in the Hankel matrix X is 1, and the rank of the second term is also 1. Based on the relationship between the sum of the ranks of two matrices, that is... Therefore, the rank of the Hankel matrix X is at most 2.

[0080] when Then, equation 23 can be transformed into the following equation:

[0081] In the formula, According to the sum-to-product formula, the rank of the Hankel matrix X in the above equation is 2.

[0082] when At that time, it can be seen from Equation 23 that the first principal minor of the Hankel matrix X is... The second-order principal minor is also not zero, that is:

[0083] It can be seen that the rank of the Hankel matrix X is at least 2.

[0084] Combining the aforementioned conclusions and the fact that the first two principal minors of the matrix are not zero, we can conclude that the rank of the Hankel matrix X is 2. Therefore, the Hankel matrix X contains two non-zero singular values. Furthermore, we can see that one frequency component generates two non-zero singular values.

[0085] Combining this with the aforementioned equivalence between singular values ​​and eigenvalues, we can conclude that each frequency component in a signal generates two non-zero singular values / effective eigenvalues.

[0086] We will further explore the quantitative relationship between the position of non-zero singular values / effective eigenvalues ​​in the eigenvalue distribution plot of the singular value distribution spectrum / covariance matrix and the amplitude of the signal frequency components.

[0087] Based on the conclusion that each frequency component of the signal generates two adjacent non-zero singular values, Equation 14 can be transformed into the following equation:

[0088] The energy of the Hankel matrix X can then be expressed as:

[0089] Based on the theory of singular value decomposition, , , , , , We can obtain:

[0090] From the way the signal is constructed into a Hankel, we can know that | X | 2 and a 2 They are directly proportional; the larger 'a' is, the greater the energy of X. and The larger the value, the higher it ranks in the singular value distribution spectrum.

[0091] Combining the aforementioned equivalence between singular values ​​and eigenvalues, and deriving from Equation 22, we can see that the order of non-zero singular values / effective eigenvalues ​​in the eigenvalue distribution map / singular value distribution spectrum of the covariance matrix is ​​determined by the amplitude of the signal frequency components. The larger the amplitude of a frequency component, the larger the non-zero singular value / effective eigenvalue it produces, and the earlier it appears in the eigenvalue distribution plot / singular value distribution spectrum of the covariance matrix. The specific relationship is as follows: , .

[0092] Secondly, there is the relationship between singular values / eigenvalues ​​and signal length: For signals of different lengths, the singular values ​​are related to the data length and the amplitude of the characteristic frequencies as follows:

[0093] in, and For a given frequency component, the singular value is... For signal length, a i This represents the amplitude corresponding to a certain frequency component.

[0094] Based on the aforementioned equivalence between singular values ​​and eigenvalues, and by deriving Equation 22, we can conclude that:

[0095] in, and The eigenvalue corresponding to a certain frequency component. Let n be the signal length and n be the number of columns in matrix X. a i This refers to the amplitude corresponding to a certain frequency component. It should be noted here that the amplitude mentioned earlier... A i The amplitude of the i-th frequency component in the signal containing multiple signal components constructed during the previous relationship summary; and here... a i This refers to the amplitude of a specific frequency component. Both refer to the amplitude of a single signal frequency component and can be considered equivalent in meaning. However, when summarizing the relationship earlier, the amplitude of the constructed single i-th signal is generally written in uppercase. A i To express.

[0096] Example 5 The following is a set of simulation examples using the amplitude filter algorithm, with the signals constructed as follows:

[0097] In Equation 33, The number of frequency components. ; The signal is Gaussian white noise with a standard deviation of 1.4, a sampling frequency of 1024Hz, and 4096 sampling points. Figure 2 As shown.

[0098] The signal is processed using an amplitude filter algorithm, and the singular value distribution is as follows: Figure 3 As shown, based on the relationship between the magnitude of the non-zero singular values ​​and the signal amplitude, the correspondence between the six larger sets of singular values ​​and the frequency components in the figure is as follows: Figure 3 The text in the document is shown.

[0099] Use the first group and the third group Singular values ​​are used to reconstruct the signal, and the result is as follows: Figure 4 As shown in (a), the reconstructed result is similar to the original signal. The results were consistent. Next, the second set of singular values ​​was used. Refactoring was performed, and the result is as follows: Figure 4 As shown in (b), the reconstructed signal has a frequency of 59 Hz and an amplitude of 0.89, which is consistent with the original signal. The frequency components are consistent. Finally, the fourth set of singular values ​​is used. Refactoring was performed, and the result is as follows: Figure 4 As shown in (c), the amplitude of the reconstructed signal is 0.614 and the frequency is 60Hz, which is consistent with the original signal. The composition is consistent. Similarly, the frequency components at 50Hz and 80Hz can be obtained through reconstruction using the fifth and sixth sets of singular values, respectively.

[0100] As can be seen, amplitude filters, as a type of amplitude-oriented filter, select a specified frequency for filtering based on the order of the characteristic frequencies in the amplitude spectrum. As long as the order of the characteristic frequencies in the amplitude spectrum is known, the specified characteristic frequency can be accurately extracted or eliminated. Even if two frequency components differ by only 1 Hz, they can be successfully separated, which is difficult for frequency-oriented filters to achieve.

[0101] Furthermore, as shown in the figure, the frequency components of the filtered signal almost completely overlap with those of the original signal, with no phase shift. Moreover, the components of the reconstructed signal are additive; extracting or eliminating a specific component is equivalent to using a subtraction operation to separate that particular component from the original signal. This linear operation does not cause phase distortion in the extracted component signal. Therefore, the amplitude filter algorithm is a zero-phase-shift frequency extraction method. This characteristic distinguishes it from wavelet filters and FIR filters, which use convolution operations for signal separation. If such filters do not possess linear phase characteristics, the separated signals will exhibit phase distortion.

[0102] Example 6 Experimental data was collected using a large rotor vibration testing bench. To monitor the rotor's operation in real time, two Kaman KD2306-1S eddy current displacement sensors were installed perpendicularly at the journals of the rotor at the near and far ends of the motor. Figure 5 As shown, data acquisition was performed using LMS Test .Lab.

[0103] Next, we analyzed one set of test data each from the near-motor end and the far-motor end of the test bench rotor under different operating conditions. The sampling frequency was 2048Hz, and the sampling points were 4096. First, we collected data at the near-motor end rotor speed of 1080 r / min (18Hz) as the first set of analysis data. Then, under different operating conditions, we collected data at the far-motor end at the same speed as the second set of analysis data. The time-domain and frequency-domain plots obtained from the above two collections are shown below. Figure 6 (a) and Figure 6 As shown in (b) of the diagram.

[0104] Next, the amplitude filter algorithm is used for analysis, and the distributions of the two eigenvalues ​​are obtained as follows: Figure 7 (a) and Figure 7 As shown in (b) above. According to Figure 7 (a) and Figure 7 From the ordering of the characteristic frequency amplitudes of the fundamental frequency 1X and the second harmonic 2X of D11 and D12 in (b), it can be seen that in the signal near the motor end, the fundamental frequency corresponds to the 1st and 2nd characteristic values, and the second harmonic corresponds to the 5th and 6th characteristic values. In the signal far from the motor end, the fundamental frequency corresponds to the 3rd and 4th characteristic values, and the second harmonic corresponds to the 1st and 2nd characteristic values. Using the two sets of characteristic signals acquired near the motor end for reconstruction, we can obtain... Figure 8 The result is shown in (a) above. Reconstructing the result using the two sets of characteristic signals acquired from the remote motor terminal yields the following: Figure 8 The result is shown in (b). It is evident that the amplitude filter algorithm has a significant effect on frequency extraction, not only filtering out noise interference but also filtering out interference from the 50Hz power frequency and other components. The extracted frequency amplitude is very close to the amplitudes of the fundamental frequency and second harmonic of the original signal, indicating that the spectral error extracted by this algorithm is negligible.

[0105] use Figure 8 (a) and Figure 8 The axis center trajectory synthesized from the filtered signal in (b) is shown below. Figure 9 (a) and Figure 9 As shown in (b) above, the synthesized shaft center trajectory after filtering the vibration signal near the motor end (Example 1) is clearly banana-shaped, while the shaft center trajectory far from the motor end (Example 2) is a typical figure-eight shape. Combining common shaft center trajectories and their corresponding causes, it can be seen that there is a slight misalignment fault near the motor end and a more severe misalignment fault far from the motor end. Therefore, it is evident that the amplitude filter algorithm has a good effect on purifying the shaft center trajectory.

[0106] Next, a harmonic wavelet algorithm was used for experimental comparison. The data from the remote motor end was purified by performing an 8-level decomposition, with a bandwidth of 4Hz. The fundamental frequency of 18Hz and the second harmonic of 36Hz fall within the frequency bands [16, 20] and [34, 38], respectively. The axis center trajectory synthesized from the characteristic frequencies after harmonic wavelet purification is shown below. Figure 9 As shown in (c) in the diagram. (and) Figure 9 Compared to (b), the amplitude filter algorithm produces a clearer purification effect, a more compact trajectory, and a more pronounced figure-eight pattern. Figure 9 The small head of the number 8 is not visible in (c). This is because the harmonic wavelet algorithm contains a large amount of noise in the frequency band when extracting feature frequencies.

[0107] In addition, wavelet packets are often used to refine the axis center trajectory. The signal from the remote motor terminal is decomposed into 8 levels using Daubechies wavelet packets (db42) with a bandwidth of 4Hz. Signals in the frequency bands containing the fundamental frequency 1X and the second harmonic 2X are extracted respectively, resulting in the filtered axis center trajectory as shown below. Figure 9 As shown in (d) in the figure, it can be seen that the axis trajectory obtained by using it is also not as good as the amplitude filter algorithm, and it is difficult to see that the axis trajectory is a figure 8 shape.

[0108] Amplitude filter algorithm to achieve single frequency filtering: The above discussion explored the application of amplitude filter algorithms in axis trajectory analysis, and then discussed the application of this invention in single-frequency filtering. Taking the experimental data of D11 as an example, since D11 contains 50Hz power frequency interference, the amplitude filter algorithm is used to filter out the 50Hz power frequency interference. Figure 7 As shown in (a), the amplitudes of the 50Hz signals in the original signal are ordered as 3. Based on the conclusion of this invention, the eigenvectors corresponding to the 5th and 6th eigenvalues ​​in the eigenvalue distribution diagram are used for reconstruction. When the amplitude filter algorithm extracts the reconstructed feature signal from the original signal, the purified signal and the original signal have an addition-subtraction relationship. Therefore, subtracting the reconstructed signal from the original data of D11 yields the signal after filtering out the 50Hz signal, and its amplitude spectrum is as follows: Figure 10 As shown.

[0109] from Figure 10 As can be seen, this algorithm not only filters out the 50Hz power frequency interference in the original signal, but also does not affect other frequency components in the original signal. This shows that this method has high application value in removing single frequencies.

[0110] The selection of effective eigenvalues ​​in the signal covariance matrix is ​​a key focus in amplitude filter algorithm research. This invention utilizes the relationship between non-zero singular values / effective eigenvalues ​​and signal frequency and amplitude, and applies the amplitude filter algorithm to the purification of axis-centered trajectories, drawing the following conclusions: (1) In the Hankel matrix approach, each effective frequency component of the signal generates at most two non-zero singular values / effective eigenvalues, closely arranged one after the other; the number of non-zero singular values / effective eigenvalues ​​is related to the number of frequencies in the signal, and... , and The values ​​are independent of the signal's amplitude; the order of the non-zero singular values / effective eigenvalues ​​in the singular value distribution spectrum / eigenvalue distribution map depends only on the amplitude corresponding to the signal frequency. Related to, with and Regardless of the value of , the larger the amplitude of the corresponding frequency, the larger the singular value / eigenvalue generated, and the earlier it is ranked.

[0111] (2) Based on the characteristics that each frequency signal generates two non-zero singular values / effective eigenvalues ​​and that the magnitude of the amplitude corresponding to the frequency determines the order of the non-zero singular values / effective eigenvalues ​​in the eigenvalue distribution map of the singular value distribution spectrum / covariance matrix, an amplitude filter algorithm is proposed. This algorithm demonstrates that even if two frequencies differ by 1 Hz, a single frequency can be extracted without being affected. This invention utilizes this characteristic to apply the amplitude filter algorithm to the purification of axisymmetric trajectories and compares its purification effect with that of wavelet packets and harmonic wavelet packets. The shortcomings of wavelet packet and harmonic wavelet packet transforms are pointed out, showing that the amplitude filter algorithm is significantly effective in purifying axisymmetric trajectories.

[0112] Example 7 The following section presents the shortcomings of the amplitude filter algorithm and corresponding solutions when the target frequency is submerged in noise. For a displacement signal of a rotating component with a rotor misalignment fault and a rotational speed of 1080 r / min, its noise component is increased so that the target feature frequency is submerged in noise. The resulting time-domain signal and amplitude spectrum are as follows: Figure 11 (a) and Figure 11 As shown in (b), both the fundamental frequency (18Hz) and the second harmonic (36Hz) signals are submerged in noise. The singular value distribution obtained under this condition is as follows: Figure 11 As shown in (c), the correspondence between each set of singular values ​​and each signal frequency cannot be known from the figure, so it is impossible to extract the features corresponding to the fundamental frequency and the second harmonic for signal reconstruction.

[0113] To solve the above problems, a certain characteristic frequency component can be added first, making the amplitude of the characteristic frequency component to be extracted different from the amplitudes of other frequency components or noise in the frequency. Then, based on the order of the new amplitude of the frequency component to be extracted in the frequency domain graph, two corresponding to the characteristic frequency to be extracted can be selected from the eigenvalue sequence spectrum of the singular value distribution spectrum / covariance matrix for singular value / eigenvalue frequency extraction. The added frequency component is then subtracted, thus completing the extraction of the characteristic frequency to be extracted. Next, a sine wave is added to the original signal to increase the amplitude of the fundamental frequency (18Hz) and second harmonic frequency (36Hz), creating a difference between them and other frequency amplitudes. The time domain signal and amplitude spectrum obtained after adding this sine wave are shown below. Figure 11 (d) in Figure 11 As shown in (e), the signal is processed using an amplitude filter algorithm, and the resulting singular value distribution is as follows. Figure 11 As shown in (f) in the figure, combined with Figure 11 The ordering of the fundamental and second harmonic characteristic frequency amplitudes of D11 and D12 in (e) shows that... Figure 11 In (f), the fundamental frequency corresponds to the 3rd and 4th singular values, and the second harmonic corresponds to the 1st and 2nd singular values. Based on the sorting, the fundamental and second harmonic frequency components are extracted, and then the added frequency components are subtracted to obtain... Figure 12 As shown in result (a), this method successfully extracts the feature components to be extracted even when the frequency components are submerged in noise. The signal after feature frequency extraction is used to synthesize the axisymmetric trajectory, and the result is as follows. Figure 12 As shown in (b) of the figure, the figure-eight-shaped axis trajectory can be clearly seen from the figure, proving the effectiveness of the solution.

[0114] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. An amplitude-guided filtering method, characterized in that, Includes the following steps: Acquire the original signal and perform filtering on it; The filtered signal is constructed in Hankel matrix form to obtain the Hankel matrix X; Based on the equivalence relationship between singular value decomposition and principal component analysis, an amplitude filter is constructed to decompose the Hankel matrix X; wherein, the equivalence relationship is predefined as: the eigenvalues ​​of the covariance matrix of the Hankel matrix X. Singular values ​​of the Hankel matrix X satisfy , n represents the column number of the Hankel matrix X, and the left singular vector obtained during singular value decomposition is equivalent to the eigenvector obtained during principal component analysis; the original signal is decomposed into multiple frequency components by an amplitude filter; Based on the amplitude spectrum of the original signal, determine the amplitude sorting position k of the target frequency component in the amplitude spectrum of the original signal; Based on the amplitude sorting position k, a set of components corresponding to the target frequency component is selected from multiple frequency components for signal reconstruction to obtain a reconstruction matrix; The time-domain signal is recovered from the reconstructed matrix, and the time-domain signal is the filtered signal of the target frequency component extracted from the original signal.

2. The amplitude-guided filtering method according to claim 1, characterized in that: The construction of the amplitude filter involves decomposing the Hankel matrix X, specifically including: Based on the aforementioned equivalence relation, singular value decomposition or principal component analysis is selected to decompose the Hankel matrix X; When using singular value decomposition, perform singular value decomposition on the Hankel matrix X to find its singular values. Arranged in descending order as follows , ,…, Construct the singular value distribution spectrum and obtain the left and right singular matrices; based on the singular values... The singular value distribution spectrum is sorted, and two singular values ​​corresponding to the target frequency component and their corresponding left and right singular vectors are selected for reconstruction. When using principal component analysis, find the covariance matrix C of the Hankel matrix X and its eigenvalues. Arranged in descending order as follows , ,…, Construct the eigenvalue distribution plot of the covariance matrix and obtain the eigenvectors corresponding to each eigenvalue; based on the eigenvalues... In the sorting of the eigenvalue distribution map of the covariance matrix, two eigenvalues ​​corresponding to the target frequency component and their corresponding eigenvectors are selected for reconstruction.

3. The amplitude-guided filtering method according to claim 2, characterized in that: Based on the sorting position k, a set of components corresponding to the target frequency component is selected for reconstruction, including: If the amplitude of a certain frequency component in the amplitude spectrum of the original signal is sorted as k, then the (2k-1)th and 2kth singular values / eigenvalues ​​and their corresponding left and right singular vectors / eigenvectors in the singular value distribution spectrum / covariance matrix eigenvalue distribution map are selected for signal reconstruction.

4. The amplitude-guided filtering method according to claim 1, characterized in that: The equivalence relation is constructed through the following steps: For the Hankel matrix X, calculate its covariance matrix. And through the covariance matrix Perform eigenvalue decomposition to obtain eigenvalues ;in: The superscript T indicates transpose; X express m × n A matrix of order 1, m Let X be the row number of the Hankel matrix. Simultaneously, singular value decomposition is performed on the Hankel matrix X to obtain singular values. : ,in, U It is a left singular matrix; V It is a right singular matrix; S It is a diagonal matrix; Based on the above relationship, the component results obtained by singular value decomposition or principal component analysis can be mapped to each other, that is, the eigenvalues ​​and singular values ​​obtained by the same Hankel matrix X under the two decompositions are consistent; the amplitude filter can select either singular value decomposition or principal component analysis for signal decomposition and reconstruction.

5. The amplitude-guided filtering method according to claim 1, characterized in that: The Hankel matrix X is constructed as follows: ； In the formula, X Let X be an m×n matrix, where m is the number of rows in the Hankel matrix X; The first element in the Hankel matrix X i Okay, number j The elements of the column are 1≤i≤m, 1≤j≤n; L is the length of the signal data, and L=m+n-1; when L is even, m=L / 2, n=L / 2+1; when L is odd, m=(L+1) / 2, n=(L+1) / 2.

6. The amplitude-guided filtering method according to claim 1, characterized in that: Recovering the time-domain signal from the reconstructed matrix includes: The average value of each diagonal element of the reconstructed matrix is ​​calculated using the averaging method to recover the one-dimensional time-domain signal; or, the inverse transformation of the reconstructed matrix is ​​performed using the pseudo-inverse matrix method to recover the one-dimensional time-domain signal.

7. The amplitude-guided filtering method according to claim 1, characterized in that: In the multiple frequency components obtained by the amplitude filter decomposition, each frequency component generates two non-zero singular values / effective eigenvalues, and the two non-zero singular values / effective eigenvalues ​​are arranged one after the other in the singular value distribution spectrum / covariance matrix eigenvalue distribution diagram. The number of non-zero singular values / effective eigenvalues ​​is only related to the number of frequency components and is independent of the amplitude, frequency, and phase of the signal. The order of non-zero singular values / effective eigenvalues ​​in the singular value distribution spectrum / covariance matrix eigenvalue distribution map is determined by the amplitude of the signal frequency components. The larger the amplitude of the frequency component, the larger the non-zero singular values / effective eigenvalues ​​it produces, and the earlier it is arranged.

8. The amplitude-guided filtering method according to claim 7, characterized in that: For signals of different lengths, the singular values ​​are related to the data length and the amplitude of the characteristic frequencies as follows: ； in, and Let L be the singular value corresponding to a certain frequency component, and L be the signal length. The amplitude corresponding to a certain frequency component; Based on the equivalence relationship between singular values ​​and eigenvalues, the above equation can be transformed as follows: ； in, and It is the eigenvalue corresponding to a certain frequency component.

9. The amplitude-guided filtering method according to claim 1, characterized in that, Also includes: When the target frequency component is submerged in noise in the amplitude spectrum of the original signal and its amplitude ranking position k cannot be determined, an auxiliary signal is injected into the original signal. The frequency of the auxiliary signal is the same as that of the target frequency component, so as to improve the amplitude ranking position of the target frequency component in the amplitude spectrum. In the new amplitude spectrum, the improved sorting position k' is determined, and the corresponding decomposition result is selected according to the improved sorting position k' for reconstruction, so as to obtain a reconstructed signal containing the target frequency component and the auxiliary signal; The auxiliary signal is subtracted from the reconstructed signal to obtain the filtered signal of the target frequency component.

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