A signal separation method for harmonic and impulse signals

Abstract

The application discloses a signal separation method for harmonic signals and impact signals, comprising the following steps: step 1, constructing a first Hankel matrix; step 2, iterative operation, updating the first Hankel matrix; step 3, calculating the estimated value of the time-domain harmonic signal according to the first Hankel matrix after alpha iteration update; step 4, subtracting the estimated value of the time-domain harmonic signal from the time-domain total signal to obtain the first estimated value of the time-domain impact signal; step 5, constructing a second Hankel matrix; step 6, iterative operation, updating the second Hankel matrix; step 7, calculating the second estimated value of the time-domain impact signal according to the first Hankel matrix after alpha iteration update, and performing inverse Fourier transform on the second estimated value of the time-domain impact signal to obtain the third estimated value of the time-domain impact signal; and step 8, subtracting the third estimated value of the time-domain impact signal from the time-domain total signal and returning to step 1.

Description

Technical Field

[0001] This invention relates to the field of signal processing technology, and in particular to a signal separation method for harmonic signals and impulse signals. Background Technology

[0002] In several key fields such as mechanical fault diagnosis, structural health monitoring, underwater acoustic signal processing, and biomedical engineering, the signals acquired by sensors are often not pure but rather a mixture of components generated by various physical processes. Among these, two of the most representative and distinct components are harmonic signals and impulse signals. Separating these two components has crucial application value, but it is also a highly challenging signal processing problem.

[0003] From an application perspective, harmonic signals typically correspond to regular, periodic motion within a system. In contrast, impulse signals represent non-periodic, transient, and energy-concentrated abrupt changes in a signal. Harmonic and impulse signals exhibit drastically different characteristics in the time and frequency domains, posing significant challenges to their separation. Harmonic signals are stationary and continuous in the time domain; in the frequency domain, their energy is concentrated at the discrete spectral lines of the fundamental frequency and its harmonics. Impulse signals, on the other hand, are sparsity and transient in the time domain, with uncertain occurrence times and short durations; in the frequency domain, due to their broadband characteristics, their energy is dispersed over a wide frequency range.

[0004] Existing separation methods mainly revolve around time-domain filtering, frequency-domain notch filtering, and time-frequency analysis, but each has significant limitations. The frequency-domain method based on the Fast Fourier Transform (FFT) is a classic approach. It calculates the signal's spectrum, identifies the spectral peaks corresponding to harmonics, and then designs a notch filter to remove energy near these peak frequencies, treating the remaining portion as impulse signals or other residues. This method is simple in principle, but impulse signals themselves are broadband in the frequency domain, and their spectrum may overlap with the sidebands or leakage spectra of harmonics. Simple frequency-domain filtering cannot completely separate the two.

[0005] In recent years, adaptive methods based on sparse signal representation and variational mode decomposition have made significant progress. Sparse representation methods assume that a signal can be sparsely represented under a certain dictionary. By constructing a harmonic dictionary (such as a Fourier basis) and an impulse dictionary (such as a pulse or wavelet packet dictionary), optimization algorithms (such as basis pursuit denoising and matched pursuit) are used to find the sparsest representation of the signal under these two dictionaries, thereby achieving separation. The performance of these methods is highly dependent on the degree of matching between the preset dictionary and the intrinsic structure of the signal. If the actual impulse pattern differs significantly from the dictionary atoms, the separation effect will be greatly reduced, and the computational complexity is usually high.

[0006] In summary, the separation of harmonic and impulse signals has wide-ranging applications in equipment operation and maintenance, structural safety assessment, underwater acoustic signal processing, and biosignal analysis. Although existing methods offer solutions at multiple levels, including frequency domain, time-frequency domain, and adaptive decomposition, they generally face problems such as dependence on prior knowledge of the signal, sensitivity to parameter selection, computational efficiency bottlenecks, and a sharp decline in separation performance at low signal-to-noise ratios.

[0007] In view of this, this invention is hereby proposed. Summary of the Invention

[0008] The purpose of this invention is to address the shortcomings of existing technologies by proposing a signal separation method for harmonic and impulse signals, which can provide technical support for signal processing in fields such as mechanical fault diagnosis and condition monitoring, biomedical signal processing, and geophysical and acoustic exploration.

[0009] To achieve the above objectives, the present invention adopts the following technical solution:

[0010] A method for separating harmonic signals and impulse signals includes the following steps:

[0011] Step 1: Receive the total time-domain signal including harmonic and impulse signals. And construct the first Hankel matrix. ;

[0012] Step 2: Iteratively execute the first Hankel matrix Singular value decomposition, singular value truncation, and anti-angle averaging operations on the first Hankel matrix. Update;

[0013] Step 3: Update the first Hankel matrix according to the αth iteration. Calculate the estimated value of the time-domain harmonic signal. ;

[0014] Step 4: Utilize the total time-domain signal Subtract the estimated value of the time-domain harmonic signal The first estimate of the time-domain impulse signal is obtained. ;

[0015] Step 5: First estimate of the time-domain impulse signal Perform a Fourier transform and construct the second Hankel matrix. ;

[0016] Step 6: Iteratively execute the second Hankel matrix Singular value decomposition, singular value truncation, and anti-angle averaging operations on the second Hankel matrix. Update;

[0017] Step 7: Update the first Hankel matrix according to the αth iteration. Calculate the second estimate of the time-domain impulse signal And the second estimate of the time-domain impact signal. The third estimate of the time-domain impulse signal is obtained by performing an inverse Fourier transform. ;

[0018] Step 8: Combine the total time-domain signal Subtract the third estimate of the time-domain impulse signal Then return to step 1 and replace the total time-domain signal with the difference. The process is iterated until convergence is achieved, yielding the final estimate of the harmonic signal. and the final estimated value of the impact signal .

[0019] Furthermore, in step 1, based on the total time-domain signal... First Hankel Matrix The format is as follows:

[0020] ;

[0021] in, The time-domain sampling interval is 2M+1, and the total length of the signal after discrete sampling is 2M+1.

[0022] Furthermore, step 2 includes the following steps:

[0023] Step 2.1: Convert the first Hankel matrix The singular value decomposition formula is as follows:

[0024] ;

[0025] ;

[0026] ;

[0027] ;

[0028] in, and It is a unitary matrix. Given a complex number domain, the total length of the discretely sampled signal is 2M+1. It is a diagonal matrix. These are singular values, i = 1 ~ M+1;

[0029] Step 2.2: For the first Hankel matrix The singular value truncation formula is as follows:

[0030] ;

[0031] ;

[0032] Where R is The number of harmonic frequencies contained therein, the number of 0s is M+1-R;

[0033] Step 2.3: For Performing an anti-angle averaging operation yields the updated first Hankel matrix. The formula is as follows:

[0034] ;

[0035] Where m and n represent respectively The row and column numbers are given by m'+n'=m+n, which represents the set of positive integer pairs whose sum is m+n.

[0036] Furthermore, in step 3, the first Hankel matrix is ​​updated based on the αth iteration. The estimated value of the time-domain harmonic signal is calculated using the following formula:

[0037] ;

[0038] Where m and n represent respectively The row and column numbers.

[0039] Furthermore, step 5 includes the following steps:

[0040] Step 5.1: First estimate of the time-domain impulse signal Performing a Fourier transform, we obtain:

[0041] ;

[0042] The total length of the discretely sampled signal is 2M+1;

[0043] Step 5.2: Utilize Constructing the second Hankel matrix The format is as follows:

[0044] ;

[0045] in, The time-domain sampling interval is 2M+1, and the total length of the signal after discrete sampling is 2M+1.

[0046] Furthermore, step 6 includes the following steps:

[0047] Step 6.1: Convert the second Hankel matrix The singular value decomposition formula is as follows:

[0048] ;

[0049] ;

[0050] ;

[0051] ;

[0052] in, and It is a unitary matrix. Given a complex number domain, the total length of the discretely sampled signal is 2M+1. It is a diagonal matrix. For example, i = 1 to M+1;

[0053] Step 6.2: For the second Hankel matrix The singular value truncation formula is as follows:

[0054] ;

[0055] ;

[0056] Where R' is the number of pulse signals in x(t), and the number of 0s is M+1-R';

[0057] Step 6.3: For Performing an anti-angle averaging operation yields the updated second Hankel matrix. The formula is as follows:

[0058] ;

[0059] Where m and n represent respectively The row and column numbers are given by m'+n'=m+n, which represents the set of positive integer pairs whose sum is m+n.

[0060] Furthermore, step 7 includes the following steps:

[0061] Step 7.1: Update the first Hankel matrix according to the αth iteration. Calculate the second estimate of the time-domain impulse signal The formula is as follows:

[0062] ;

[0063] Where m and n represent respectively Row and column numbers;

[0064] Step 7.2: Second estimate of the time-domain impulse signal The third estimate of the time-domain impulse signal is obtained by performing an inverse Fourier transform. The formula is as follows:

[0065] .

[0066] Compared with the prior art, the beneficial effects of this invention are as follows:

[0067] By utilizing the sparsity of harmonic signals in the frequency domain and the sparsity of impulse signals in the time domain, effective separation of harmonic and impulse signals is achieved through alternating projection in the time and frequency domains, effectively avoiding contamination of harmonic signals by broadband impulse signals. This method can be applied to fields such as mechanical fault diagnosis and condition monitoring, biomedical signal processing, and geophysical and acoustic exploration. Attached Figure Description

[0068] Figure 1 A flowchart of a signal separation method for harmonic signals and impulse signals;

[0069] Figure 2 for A curve graph;

[0070] Figure 3 for A curve graph;

[0071] Figure 4 for A curve graph;

[0072] Figure 5 for A curve graph;

[0073] Figure 6 for The curve graph. Detailed Implementation

[0074] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0075] Example 1: A signal separation method for harmonic signals and impulse signals, comprising the following steps:

[0076] Step 1: Receive the total time-domain signal including harmonic and impulse signals. And construct the first Hankel matrix. .

[0077] In this embodiment, in step 1, based on the total time-domain signal... The first Hankel matrix constructed The format is as follows:

[0078] ;

[0079] in, The time-domain sampling interval is 2M+1, and the total length of the signal after discrete sampling is 2M+1.

[0080] Step 2: Iteratively execute the first Hankel matrix Singular value decomposition, singular value truncation, and anti-angle averaging operations on the first Hankel matrix. Update.

[0081] In this embodiment, step 2 includes the following steps:

[0082] Step 2.1: Convert the first Hankel matrix The singular value decomposition formula is as follows:

[0083] ;

[0084] ;

[0085] ;

[0086] ;

[0087] in, and It is a unitary matrix. Given a complex number domain, the total length of the discretely sampled signal is 2M+1. It is a diagonal matrix. For singular values, i = 1 ~ M+1.

[0088] Step 2.2: For the first Hankel matrix The singular value truncation formula is as follows:

[0089] ;

[0090] ;

[0091] Where R is The number of harmonic frequencies contained therein is M+1-R, and the number of 0s is M+1-R.

[0092] Step 2.3: For Performing an anti-angle averaging operation yields the updated first Hankel matrix. The formula is as follows:

[0093] ;

[0094] Where m and n represent respectively The row and column numbers are given by m'+n'=m+n, which represents the set of positive integer pairs whose sum is m+n.

[0095] Step 3: Update the first Hankel matrix according to the αth iteration. Calculate the estimated value of the time-domain harmonic signal. .

[0096] In this embodiment, in step 3, the first Hankel matrix is ​​updated according to the αth iteration. The estimated value of the time-domain harmonic signal is calculated using the following formula:

[0097] ;

[0098] Where m and n represent respectively The row and column numbers.

[0099] Step 4: Utilize the total time-domain signal Subtract the estimated value of the time-domain harmonic signal The first estimate of the time-domain impulse signal is obtained. .

[0100] Step 5: First estimate of the time-domain impulse signal Perform a Fourier transform and construct the second Hankel matrix. .

[0101] In this embodiment, step 5 includes the following steps:

[0102] Step 5.1: First estimate of the time-domain impulse signal Performing a Fourier transform, we obtain:

[0103] ;

[0104] The total length of the discretely sampled signal is 2M+1;

[0105] Step 5.2: Utilize Constructing the second Hankel matrix The format is as follows:

[0106] ;

[0107] in, The time-domain sampling interval is 2M+1, and the total length of the signal after discrete sampling is 2M+1.

[0108] Step 6: Iteratively execute the second Hankel matrix Singular value decomposition, singular value truncation, and anti-angle averaging operations on the second Hankel matrix. Update.

[0109] In this embodiment, step 6 includes the following steps:

[0110] Step 6.1: Convert the second Hankel matrix The singular value decomposition formula is as follows:

[0111] ;

[0112] ;

[0113] ;

[0114] ;

[0115] in, and It is a unitary matrix. Given a complex number domain, the total length of the discretely sampled signal is 2M+1. It is a diagonal matrix. Let i = 1 ~ M+1.

[0116] Step 6.2: For the second Hankel matrix The singular value truncation formula is as follows:

[0117] ;

[0118] ;

[0119] Where R' is the number of pulse signals in x(t), and the number of 0s is M+1-R';

[0120] Step 6.3: For Performing an anti-angle averaging operation yields the updated second Hankel matrix. The formula is as follows:

[0121] ;

[0122] Where m and n represent respectively The row and column numbers are given by m'+n'=m+n, which represents the set of positive integer pairs whose sum is m+n.

[0123] Based on the first Hankel matrix updated after α iterations Calculate the second estimate of the time-domain impulse signal And the second estimate of the time-domain impulse signal. The third estimate of the time-domain impulse signal is obtained by performing an inverse Fourier transform. .

[0124] In this embodiment, step 7 includes the following steps:

[0125] Step 7.1: Update the first Hankel matrix according to the αth iteration. Calculate the second estimate of the time-domain impulse signal The formula is as follows:

[0126] ;

[0127] Where m and n represent respectively The row and column numbers.

[0128] Step 7.2: Second estimate of the time-domain impulse signal The third estimate of the time-domain impulse signal is obtained by performing an inverse Fourier transform. The formula is as follows:

[0129] .

[0130] Step 8: Combine the total time-domain signal Subtract the third estimate of the time-domain impulse signal Then return to step 1 and replace the total time-domain signal with the difference. The process is iterated until convergence is achieved, yielding the final estimate of the harmonic signal. and the final estimated value of the impact signal .

[0131] This embodiment presents a signal separation method for harmonic and impulse signals. It utilizes the sparsity of harmonic signals in the frequency domain and the sparsity of impulse signals in the time domain, achieving effective separation of harmonic and impulse signals through alternating projection into the time and frequency domains. This effectively avoids contamination of harmonic signals by broadband impulse signals. It can be applied to fields such as mechanical fault diagnosis and condition monitoring, biomedical signal processing, and geophysical and acoustic exploration.

[0132] To verify the effectiveness of the method in this embodiment, a frequency-modulated underwater acoustic communication signal affected by impulse noise was used as an example for verification. Figure 2 As shown, its total time-domain signal The format is as follows:

[0133] ;

[0134] Among them, c1=1.619+0.484i, c2=0.055+1.598i, c3=0.765+0.688i, c4=-0.822+0.652i, c5=1.254+0.161i, f1=18.072, f2=44.677, f3=90.5 54, f4=33.064, f5=7.139, b1=5.258, b2=8.553, b3=7.129, b4=6.028, b5=7.534, t1=0.465, t2=0.312, t3=0.046, t4=0.085, t5=0.235.

[0135] Based on the total time domain signal The first Hankel matrix constructed The format is as follows:

[0136] ;

[0137] in, M=249.

[0138] The first Hankel matrix The singular value decomposition formula is as follows:

[0139] ;

[0140] ;

[0141] ;

[0142] ;

[0143] For the first Hankel matrix The singular value truncation formula is as follows:

[0144] ;

[0145] ;

[0146] right Performing an anti-angle averaging operation yields the updated first Hankel matrix. The formula is as follows:

[0147] .

[0148] Repeat the first Hankel matrix The singular value decomposition, singular value truncation, and anti-angle averaging operations are performed 10 times on the first Hankel matrix. Update.

[0149] Based on the first Hankel matrix after 10 iterations and updates The estimated value of the time-domain harmonic signal is calculated using the following formula:

[0150] .

[0151] Using the total time-domain signal Subtract the estimated value of the time-domain harmonic signal The first estimate of the time-domain impulse signal is obtained. .

[0152] First estimate of the time-domain impulse signal Performing a Fourier transform, we obtain:

[0153] .

[0154] use Constructing the second Hankel matrix The format is as follows:

[0155] ;

[0156] in, The time-domain sampling interval is M=249.

[0157] The second Hankel matrix The singular value decomposition formula is as follows:

[0158] ;

[0159] ;

[0160] ;

[0161] ;

[0162] For the second Hankel matrix The singular value truncation formula is as follows:

[0163] ;

[0164] ;

[0165] right Performing an anti-angle averaging operation yields the updated second Hankel matrix. The formula is as follows:

[0166] ;

[0167] Repeating the second Hankel matrix The singular value decomposition, singular value truncation, and anti-angle averaging operations are performed 10 times on the second Hankel matrix. Update.

[0168] Based on the first Hankel matrix after 10 iterations and updates Calculate the second estimate of the time-domain impulse signal The formula is as follows:

[0169] .

[0170] Second estimate of the time-domain impulse signal The third estimate of the time-domain impulse signal is obtained by performing an inverse Fourier transform. The formula is as follows:

[0171] .

[0172] The total time-domain signal Subtract the third estimate of the time-domain impulse signal Then return to step 1 and replace the total time-domain signal with the difference. After 10 iterations, the final estimate of the harmonic signal is obtained. and the final estimated value of the impact signal .

[0173] After only one iteration, The results are as follows Figure 3 As shown, it can be seen and Significant differences exist, with the largest difference in signal amplitude reaching 2, approximately one-third of the earlier signal amplitude. After two iterations... The results are as follows Figure 4 As shown in the figure, it can be seen that after two iterations, the position of the pulse signal can be accurately obtained, but the amplitude estimation error of the pulse signal is relatively large. The results after 10 iterations are as follows: Figure 5 and 6 As shown, it can be seen that after 10 iterations, and With almost no difference, after 10 iterations, not only can the position of the pulse signal be accurately obtained, but the amplitude of the pulse signal can also be accurately estimated, thus successfully separating the harmonic signal and the impulse signal.

[0174] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A signal separation method for harmonic signals and impulse signals, characterized in that, Includes the following steps: Step 1: Receive the total time-domain signal including harmonic and impulse signals. And construct the first Hankel matrix. ; Step 2: Iteratively execute the first Hankel matrix Singular value decomposition, singular value truncation, and anti-angle averaging operations on the first Hankel matrix. Update; Step 3: Update the first Hankel matrix according to the αth iteration. Calculate the estimated value of the time-domain harmonic signal. ; Step 4: Utilize the total time-domain signal Subtract the estimated value of the time-domain harmonic signal The first estimate of the time-domain impulse signal is obtained. ; Step 5: First estimate of the time-domain impulse signal Perform a Fourier transform and construct the second Hankel matrix. ; Step 6: Iteratively execute the second Hankel matrix Singular value decomposition, singular value truncation, and anti-angle averaging operations on the second Hankel matrix. Update; Step 7: Update the first Hankel matrix according to the αth iteration. Calculate the second estimate of the time-domain impulse signal And the second estimate of the time-domain impulse signal. The third estimate of the time-domain impulse signal is obtained by performing an inverse Fourier transform. ; Step 8: Combine the total time-domain signal Subtract the third estimate of the time-domain impulse signal Then return to step 1 and replace the total time-domain signal with the difference. The process is iterated until convergence is achieved, yielding the final estimate of the harmonic signal. and the final estimated value of the impact signal .

2. The signal separation method for harmonic signals and impulse signals according to claim 1, characterized in that, In step 1, based on the total time-domain signal First Hankel Matrix The format is as follows: ； in, The time-domain sampling interval is 2M+1, and the total length of the signal after discrete sampling is 2M+1.

3. The signal separation method for harmonic signals and impulse signals according to claim 2, characterized in that, Step 2 includes the following steps: Step 2.1: Convert the first Hankel matrix The singular value decomposition formula is as follows: ； ； ； ； in, and It is a unitary matrix. Given a complex number domain, the total length of the discretely sampled signal is 2M+1. It is a diagonal matrix. These are singular values, i = 1 ~ M+1; Step 2.2: For the first Hankel matrix The singular value truncation formula is as follows: ； ； Where R is The number of harmonic frequencies contained therein, the number of 0s is M+1-R; Step 2.3: For Performing an anti-angle averaging operation yields the updated first Hankel matrix. The formula is as follows: ； Where m and n represent respectively The row and column numbers are given by m'+n'=m+n, which represents the set of positive integer pairs whose sum is m+n.

4. The signal separation method for harmonic signals and impulse signals according to claim 2, characterized in that, In step 3, the first Hankel matrix is ​​updated based on the αth iteration. The estimated value of the time-domain harmonic signal is calculated using the following formula: ； Where m and n represent respectively The row and column numbers.

5. The signal separation method for harmonic signals and impulse signals according to claim 1, characterized in that, Step 5 includes the following steps: Step 5.1: First estimate of the time-domain impulse signal Performing a Fourier transform, we obtain: ； The total length of the discretely sampled signal is 2M+1; Step 5.2: Utilize Constructing the second Hankel matrix The format is as follows: ； in, The time-domain sampling interval is 2M+1, and the total length of the signal after discrete sampling is 2M+1.

6. The signal separation method for harmonic signals and impulse signals according to claim 5, characterized in that, Step 6 includes the following steps: Step 6.1: Convert the second Hankel matrix The singular value decomposition formula is as follows: ； ； ； ； in, and It is a unitary matrix. Given a complex number domain, the total length of the discretely sampled signal is 2M+1. It is a diagonal matrix. For example, i = 1 to M+1; Step 6.2: For the second Hankel matrix The singular value truncation formula is as follows: ； ； Where R' is the number of pulse signals in x(t), and the number of 0s is M+1-R'; Step 6.3: For Performing an anti-angle averaging operation yields the updated second Hankel matrix. The formula is as follows: ； Where m and n represent respectively The row and column numbers are given by m'+n'=m+n, which represents the set of positive integer pairs whose sum is m+n.

7. The signal separation method for harmonic signals and impulse signals according to claim 5, characterized in that, Step 7 includes the following steps: Step 7.1: Update the first Hankel matrix according to the αth iteration. Calculate the second estimate of the time-domain impulse signal The formula is as follows: ； Where m and n represent respectively Row and column numbers; Step 7.2: Second estimate of the time-domain impulse signal The third estimate of the time-domain impulse signal is obtained by performing an inverse Fourier transform. The formula is as follows: 。

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