Power distribution network signal sparse decomposition method based on combined over-complete dictionary

Abstract

The invention discloses a power distribution network signal sparse decomposition method based on a combined over-complete dictionary, and belongs to the technical field of power systems. The invention discloses a power distribution network signal sparse decomposition method based on a combined over-complete dictionary. The method comprises the following steps: carrying out time-frequency characteristic analysis on traditional Gabor over-complete dictionary atoms; expansion atoms suitable for periodic signal decomposition in a window are expanded on a traditional Gabor over-complete dictionary, and an Ex-Gabor over-complete dictionary is designed; time-frequency analysis is carried out on the Damped-sine over-complete dictionary; and the Ex-Gabor over-complete dictionary and the Damped-sine over-complete dictionary are combined to be used for signal sparse decomposition. By adopting the power distribution network signal sparse decomposition method based on the combined over-complete dictionary, the problems of low signal decomposition precision and low efficiency caused by relatively high signal complexity and relatively large noise interference in a power distribution network can be solved.

Description

Technical Field

[0001] The present invention relates to the technical field of power systems, and in particular to a sparse decomposition method for distribution network signals based on a combined overcomplete dictionary. Background Art

[0002] With the large-scale integration of distributed renewable energy, the operation of new distribution systems has become increasingly complex, posing new challenges to the control and protection of distribution networks. This requires real-time, comprehensive monitoring of distribution networks to provide a data foundation for upgrading control and protection solutions. In distribution networks, due to the widespread distribution of measurement points and poor communication environments, the data upload frequency of each phasor measurement unit (PMU) is limited. Effectively monitoring phasor data fluctuations within limited resources is key to achieving comprehensive synchronized phasor measurement in distribution networks. Furthermore, PMU measurement results cannot accurately describe the characteristics of transient fault signals. Therefore, real-time waveform measurement is essential to obtain comprehensive information about the fault process.

[0003] Sparse decomposition theory, a recently developed signal analysis and processing technique, has garnered widespread attention for its efficient and precise signal representation. Its primary advantage lies in its ability to represent complex signals as a combination of a finite number of basis functions, enabling efficient feature extraction, noise suppression, and extraction of key information. In electrical engineering, sparse techniques are widely used in fault location, stability assessment, and dynamic monitoring.

[0004] The main processes of sparse decomposition theory include overcomplete dictionary design and matching pursuit. Overcomplete dictionary design is the first step in sparse decomposition. The purpose is to represent the signal by selecting a set of basis functions (or atoms). These basis functions have more dimensions than the signal, thereby providing more representation flexibility. Distribution network signals are periodic, volatile, and sudden. A single overcomplete dictionary cannot efficiently match all features. Therefore, the present invention aims to combine the characteristics of distribution network measurement signals and propose a combined overcomplete dictionary design method suitable for distribution network signal decomposition. Summary of the Invention

[0005] The purpose of the present invention is to provide a sparse decomposition method for distribution network signals based on a combined overcomplete dictionary to solve the problems of low signal decomposition accuracy and low efficiency caused by high signal complexity and large noise interference in the distribution network.

[0006] To achieve the above object, the present invention provides a method for sparse decomposition of distribution network signals based on a combined overcomplete dictionary, comprising the following steps:

[0007] S1. Analyze the time-frequency characteristics of the traditional Gabor overcomplete dictionary atoms to determine their applicability to decomposing fluctuating signals within the window.

[0008] S2. Expand the traditional Gabor overcomplete dictionary with extended atoms suitable for signal decomposition with periodicity within the window and design the Ex-Gabor overcomplete dictionary.

[0009] S3, perform time-frequency analysis on the Damped-sine overcomplete dictionary to determine its use in decomposing signals with mutations within the window;

[0010] S4. The Ex-Gabor overcomplete dictionary and the Damped-sine overcomplete dictionary are combined for signal sparse decomposition, and a combined overcomplete dictionary retrieval strategy combining the distribution network signal characteristics is proposed.

[0011] Preferably, in S1, the time-frequency characteristic analysis of the traditional Gabor overcomplete dictionary atoms includes the following steps:

[0012] S11, sparse decomposition of signals in distribution network based on traditional Gabor overcomplete dictionary;

[0013] For M atoms d i The overcomplete dictionary D is composed of N atomic signals, and the input signal s is decomposed by iteratively selecting the atom d that best matches the current signal in the overcomplete dictionary D. i ; In the yth iteration, reconstruct the signal s y And the residual signal R(s) are:

[0014]

[0015] Among them, α i It is atom d i The sparse decomposition coefficient of ;

[0016] The residual signal R(s) and the atom d in the overcomplete dictionary i The atom with the largest absolute value of the inner product is the best matching atom.

[0017]

[0018] Among them, <> represents the inner product, i y is the index of the best matching atom;

[0019] The mathematical expression of S12 and Gabor atoms is:

[0020]

[0021] in, represents the Gaussian window function; γ1 is a parameter group, γ1 = (s1, λ1, μ1, ω1), s1, λ1, μ1 and ω1 are the scaling, translation, modulation frequency and phase of the Gabor atom respectively; t represents time;

[0022] Discrete s1, λ1, μ1 and ω1 in equation (3), and the expressions are as follows:

[0023]

[0024] Where a = 2, Δλ = 1 / 2, Δμ = π, Δφ = π / 6; j, r, k and i are all adjustment parameters, and (j, r, k, i) ∈ Z 4 , Z 4 is a set of integers, satisfying the following constraints:

[0025]

[0026] Preferably, in S2, the process of forming the expanded new atoms includes:

[0027] S21. Replace w(t) in formula (3) with a constant A to generate an extended atom with periodicity in the entire data window. The expression of the extended atom is:

[0028]

[0029] where s2, μ2, and ω2 are the scaling, modulation frequency, and phase of the new atom, respectively;

[0030] S22, discretize s2, μ2 and ω2 in equation (6), and the expression is as follows:

[0031]

[0032] Where a = 2, Δλ = 1 / 2, Δμ = π, Δφ = π / 6; j, k and i are all adjustment parameters, and (j, r, i)∈Z 4 , Z 4 is a set of integers, satisfying the following constraints:

[0033]

[0034] Preferably, in S3, performing time-frequency analysis on the Damped-sine overcomplete dictionary includes the following steps:

[0035] The mathematical expression of S31 and Damped-sine atoms is:

[0036]

[0037] Where H(t) is a step function and satisfies the following expression:

[0038]

[0039] Where s3, λ3, μ3 and ω3 are the scale, time center, frequency center and phase of the Damped-sine atom respectively;

[0040] S32, discretize s3, λ3, μ3 and ω3 in equation (9), and the expression is as follows:

[0041]

[0042] Where a = 2, Δλ = 1 / 2, Δμ = π, Δφ = π / 6; j, r, k and i are all adjustment parameters, and (j, r, k, i) ∈ Z 4 , Z 4 is a set of integers, satisfying the following constraints:

[0043]

[0044] Preferably, in S4, the specific process of combining the Ex-Gabor overcomplete dictionary and the Damped-sine overcomplete dictionary for retrieval is as follows:

[0045] S41. First, the overall outline of the signal to be recovered is determined by searching among the extended atoms of the Ex-Gabor overcomplete dictionary in equation (6). Then, matching is performed among the traditional Gabor atoms in equation (3) to correct the reconstruction error caused by local fluctuations. The matching process is:

[0046]

[0047] Where p is the number of iterations of the matching pursuit process, p ini is the upper limit of the number of iterations to search in the expanded atoms;

[0048] S42. After reconstructing the periodic features and local fluctuation features of the signal based on the Ex-Gabor overcomplete dictionary, determine whether there is a mutation and oscillation in the signal; if the maximum value of the residual signal in formula (1) is less than the set value, it is determined that there is no mutation in the signal; if the maximum value of the residual signal in formula (1) is greater than or equal to the set value, it is determined that there is a mutation in the signal, and the mutation signal is reconstructed by searching for atoms in the Damped-sine overcomplete dictionary; this process is expressed as:

[0049]

[0050] Where, ξ th is the threshold, usually set to 20; r max is the maximum value of the residual signal R(s) in formula (1), r aver is the average value of the residual signal during the iterative process.

[0051] The advantages and positive effects of the distribution network signal sparse decomposition method based on combined overcomplete dictionary described in the present invention are:

[0052] 1. Using the traditional Gabor overcomplete dictionary to decompose fluctuating signals, the Ex-Gabor overcomplete dictionary to process periodic signals, and the Damped-sine overcomplete dictionary to deal with sudden changes in signals, the combined use can comprehensively and accurately characterize the characteristics of distribution network signals and adapt to complex signal characteristics.

[0053] 2. In the combined overcomplete dictionary retrieval strategy, the initial matching stage searches among the extended atoms of the Ex-Gabor overcomplete dictionary, which can quickly determine the overall outline of the signal to be recovered. Traditional Gabor atoms are then used to correct the reconstruction errors caused by local fluctuations. This global-first-local-then approach strikes a balance between global periodicity and local fluctuations, improving the efficiency of matching pursuit.

[0054] 3. After reconstructing the signal based on the Ex-Gabor overcomplete dictionary, the residual size is compared to determine whether there are mutations and oscillations in the signal. When the residuals caused by mutations and oscillations are much larger than the average residual, the Damped-sine overcomplete dictionary is used to reconstruct these features. Damped-sine atoms have excellent time and frequency positioning capabilities and can simultaneously capture frequency changes and energy attenuation. They can accurately reconstruct transient or attenuated signals, thereby more accurately restoring the true state of the signal and improving signal reconstruction accuracy.

[0055] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 is a flow chart of an embodiment of the present invention;

[0057] Figure 2 This is a schematic diagram of the sparse decomposition process according to an embodiment of the present invention;

[0058] Figure 3 This is a traditional Gabor atom time-frequency decomposition diagram of an embodiment of the present invention;

[0059] Figure 4 Result diagram of the real and imaginary parts of the phasor containing interharmonics and local fluctuations according to an embodiment of the present invention;

[0060] Figure 5 This is a time-frequency decomposition diagram of the Damped-sine atom according to an embodiment of the present invention.

[0061] Figure 6 The on-site waveform recording data containing mutations and oscillations according to an embodiment of the present invention;

[0062] Figure 7This is a combined over-complete dictionary retrieval process for on-site metallic grounding fault recording data of the present invention. DETAILED DESCRIPTION

[0063] In the description of the present invention, it should be noted that the terms "upper", "lower", "inside", "outside" and the like indicate orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, or the orientations or positional relationships in which the inventive product is usually placed when in use. These are only for the convenience of describing the present invention and simplifying the description, and are not intended to indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation. Therefore, they cannot be understood as limitations on the present invention. In the description of the present invention, it should also be noted that, unless otherwise expressly specified and limited, the terms "setting", "installation" and "connection" should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection, or an indirect connection through an intermediate medium, or it can be a communication between the internal parts of two elements. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0064] In this application, unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art to which this application belongs. In the event of any inconsistency, the meaning described in this specification or the meaning derived from the contents recorded in this specification shall prevail. In addition, the terms used herein are only for the purpose of describing the embodiments of this application and are not intended to limit this application.

[0065] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0066] like Figure 1 、 Figure 2 A sparse decomposition method for distribution network signals based on a combined overcomplete dictionary includes the following steps:

[0067] S1. Analyze the time-frequency characteristics of the traditional Gabor overcomplete dictionary atoms to determine their applicability to decomposing fluctuating signals within the window.

[0068] The time-frequency characteristics analysis of traditional Gabor overcomplete dictionary atoms includes the following steps:

[0069] S11. Based on the traditional Gabor overcomplete dictionary, the signal in the distribution network is sparsely decomposed, with the aim of using a linear combination of a small number of atoms to approximate the original signal.

[0070] The design process of an overcomplete dictionary is to construct atomic signals based on signal characteristics, and then provide a dictionary search strategy to identify the atomic linear combination that can most accurately represent the original signal.

[0071] For M atoms d i The matching pursuit process involves iteratively selecting the atom d in the overcomplete dictionary D that best matches the current signal. i , whose goal is to gradually approximate the signal s. In the yth iteration, the reconstructed signal s y And the residual signal R(s) are:

[0072]

[0073] Among them, α i It is atom d i The sparse decomposition coefficient of ;

[0074] The iterative process of selecting the best matching atom each time can be regarded as a global optimization problem. i The atom with the largest absolute value of the inner product is the best matching atom.

[0075]

[0076] Among them, <> represents the inner product, i y is the index of the best matching atom.

[0077] Due to the harsh operating environment of distribution networks, signals are complex and highly variable, making it difficult to efficiently match all features using a single overcomplete dictionary. The Gabor overcomplete dictionary is a commonly used overcomplete atom library for decomposing time-varying signals, but it also has its limitations. Therefore, it is necessary to design a combined overcomplete dictionary based on the specific characteristics of the signal.

[0078] The mathematical expression of S12 and Gabor atoms is:

[0079]

[0080] in, represents the Gaussian window function; γ1 is a parameter group, γ1 = (s1, λ1, μ1, ω1), s1, λ1, μ1 and ω1 are the scaling, translation, modulation frequency and phase of the Gabor atom respectively; t represents time;

[0081] Discrete s1, λ1, μ1 and ω1 in equation (3), and the expressions are as follows:

[0082]

[0083] Where a = 2, Δλ = 1 / 2, Δμ = π, Δφ = π / 6; j, r, k and i are all adjustment parameters, and (j, r, k, i) ∈ Z 4 , Z 4is a set of integers, satisfying the following constraints:

[0084]

[0085] like Figure 3 、 Figure 4 As shown in the figure. From the time-frequency diagram of the Gabor atom, it can be seen that through the combination of the Gaussian window function and the sine function, the energy can be highly concentrated in a certain time interval and frequency interval. For complex signals, the Gabor transform can identify the energy changes of different frequency components at different time points, thereby revealing the instantaneous frequency and local characteristics of the signal. The real and imaginary parts of the phasor measurement results of the signal containing a certain amount of interharmonics show local fluctuations. Combined with the time domain expression and time-frequency analysis diagram of the traditional Gabor atom, due to its multi-scale and multi-directional characteristics, the Gabor dictionary is very suitable for local feature matching, and is therefore very suitable for representing signals with strong local fluctuations.

[0086] S2. Expand the traditional Gabor overcomplete dictionary with extended atoms suitable for signal decomposition with periodicity within the window and design the Ex-Gabor overcomplete dictionary.

[0087] The process of forming new atoms includes:

[0088] S21. In distribution networks, signals are time-varying and complex, but in most cases they are periodic. However, for periodic signals within a data window, traditional Gabor overcomplete dictionaries treat this periodicity as multiple local features and match them separately, resulting in inefficient matching pursuit and low reconstruction accuracy.

[0089] In formula (3), the window function w(t) limits the periodicity of the cosine function to a local area. Therefore, replacing w(t) in formula (3) with a constant A generates an extended atom with periodicity in the entire data window. The expression of the extended atom is:

[0090]

[0091] where s2, μ2, and ω2 are the scaling, modulation frequency, and phase of the new atom, respectively.

[0092] S22, discretize s2, μ2 and ω2 in equation (6), and the expression is as follows:

[0093]

[0094] Where a = 2, Δλ = 1 / 2, Δμ = π, Δφ = π / 6; j, k and i are all adjustment parameters, and (j, k, i)∈Z 4 , Z 4 is a set of integers, satisfying the following constraints:

[0095]

[0096] S3. Perform time-frequency analysis on the Damped-sine overcomplete dictionary to determine its applicability to decomposition of signals with mutations within the window.

[0097] Time-frequency analysis of the Damped-sine overcomplete dictionary includes the following steps:

[0098] The mathematical expression of S31 and Damped-sine atoms is:

[0099]

[0100] Where H(t) is a step function and satisfies the following expression:

[0101]

[0102] where s3, λ3, μ3, and ω3 are the scale, time center, frequency center, and phase of the Damped-sine atom, respectively.

[0103] S32, discretize s3, λ3, μ3 and ω3 in equation (9), and the expression is as follows:

[0104]

[0105] Where a = 2, Δλ = 1 / 2, Δμ = π, Δφ = π / 6; j, r, k and i are all adjustment parameters, and (j, r, k, i) ∈ Z 4 , Z 4 is a set of integers, satisfying the following constraints:

[0106]

[0107] like Figure 5 、 Figure 6 As shown in the figure, the time-frequency diagram of the Damped-sine atom accurately displays the details of the instantaneous frequency changes and energy decay in the signal. Its energy distribution on the time axis is progressive, meaning that over time, the energy decreases and expands toward the low-frequency region. Transient fault waveforms in distribution networks can exhibit sudden changes and oscillatory behavior. Combined with the time-domain expression and time-frequency analysis diagram of the Damped-sine atom, it possesses excellent time and frequency localization capabilities, capable of simultaneously capturing frequency changes and energy decay, making it ideal for representing transient or decaying signals.

[0108] S4. The Ex-Gabor overcomplete dictionary and the Damped-sine overcomplete dictionary are combined for signal sparse decomposition, and a combined overcomplete dictionary retrieval strategy combining the distribution network signal characteristics is proposed.

[0109] The specific process of combining the Ex-Gabor overcomplete dictionary and the Damped-sine overcomplete dictionary for retrieval is as follows:

[0110] S41. In order to improve the efficiency of matching pursuit, it is necessary to find a balance between global periodicity and local fluctuations.

[0111] First, the overall outline of the signal to be recovered is determined by searching among the extended atoms of the Ex-Gabor overcomplete dictionary in equation (6). Then, matching is performed among the traditional Gabor atoms in equation (3) to correct the reconstruction error caused by local fluctuations; the matching process is:

[0112]

[0113] Where p is the number of iterations of the matching pursuit process; p ini It is the upper limit of the number of iterations to search in the expanded atoms, which is generally 10% of the total number.

[0114] S42. After reconstructing the periodic characteristics and local fluctuation characteristics of the signal based on the Ex-Gabor overcomplete dictionary, the residual caused by mutation and oscillation will be much larger than the average residual in the reconstruction process.

[0115] Therefore, it is possible to determine whether there is a mutation and oscillation in the signal. If the maximum value of the residual signal in formula (1) is less than the set value, it is determined that there is no mutation in the signal. If the maximum value of the residual signal in formula (1) is greater than or equal to the set value, it is determined that there is a mutation in the signal, and the mutation signal is reconstructed by searching the atoms of the Damped-sine overcomplete dictionary; the process is expressed as:

[0116]

[0117] Where, ξ th is the threshold, usually set to 20; r max is the maximum value of the residual signal R(s) in formula (1), r aver is the average value of the residual signal during the iterative process.

[0118] like Figure 7 Taking the reconstruction process of on-site metallic grounding fault recording data as an example, the overall retrieval method of the designed combined over-complete dictionary is demonstrated.

[0119] In the initial stage, when the number of matching iterations is less than p iniWhen , we first search for the extended atoms defined in (6) to reconstruct the overall profile of the signal.

[0120] Then, when the number of iterations exceeds p ini When , the original atoms of the Gabor overcomplete dictionary are searched to reconstruct the local fluctuations of the signal.

[0121] Finally, it is determined whether there is a mutation in the signal. For mutations in the signal, reconstruction is performed by searching for atoms from the overcomplete Damped-sine dictionary.

[0122] Therefore, the sparse decomposition method of distribution network signals based on combined overcomplete dictionaries described in the present invention can solve the problems of low signal decomposition accuracy and low efficiency caused by high signal complexity and large noise interference in the distribution network.

[0123] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit the same. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solutions of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A sparse decomposition method for distribution network signals based on a combined overcomplete dictionary, characterized in that: The following steps are involved: S1. Analyze the time-frequency characteristics of the traditional Gabor overcomplete dictionary atoms to determine their applicability to decomposing fluctuating signals within the window. S2. Expand the traditional Gabor overcomplete dictionary with extended atoms suitable for signal decomposition with periodicity within the window and design the Ex-Gabor overcomplete dictionary. S3, perform time-frequency analysis on the Damped-sine overcomplete dictionary to determine its use in decomposing signals with mutations within the window; S4. The Ex-Gabor overcomplete dictionary and the Damped-sine overcomplete dictionary are combined for signal sparse decomposition, and a combined overcomplete dictionary retrieval strategy combining the distribution network signal characteristics is proposed.

2. The method for sparse decomposition of distribution network signals based on a combined overcomplete dictionary according to claim 1, characterized in that: In S1, the time-frequency characteristic analysis of the traditional Gabor overcomplete dictionary atoms includes the following steps: S11, sparse decomposition of signals in distribution network based on traditional Gabor overcomplete dictionary; For M atoms d i The overcomplete dictionary D is composed of N atomic signals, and the input signal s is decomposed by iteratively selecting the atom d that best matches the current signal in the overcomplete dictionary D. i ; In the yth iteration, reconstruct the signal s y And the residual signal R(s) are: Among them, α i It is atom d i The sparse decomposition coefficient of ; The residual signal R(s) and the atom d in the overcomplete dictionary i The atom with the largest absolute value of the inner product is the best matching atom. Among them, <> represents the inner product, i y is the index of the best matching atom; The mathematical expression of S12 and Gabor atoms is: in, represents the Gaussian window function; γ1 is a parameter group, γ1 = (s1, λ1, μ1, ω1), s1, λ1, μ1 and ω1 are the scaling, translation, modulation frequency and phase of the Gabor atom respectively; t represents time; Discrete s1, λ1, μ1 and ω1 in equation (3), and the expressions are as follows: Where a = 2, Δλ = 1 / 2, Δμ = π, Δφ = π / 6; j, r, k and i are all adjustment parameters, and (j, r, k, i) ∈ Z 4 , Z 4 is a set of integers, satisfying the following constraints:

3. The method for sparse decomposition of distribution network signals based on a combined overcomplete dictionary according to claim 2, characterized in that: In S2, the formation process of the expanded new atoms includes: S21. Replace w(t) in formula (3) with a constant A to generate an extended atom with periodicity in the entire data window. The expression of the extended atom is: where s2, μ2, and ω2 are the scaling, modulation frequency, and phase of the new atom, respectively; S22, discretize s2, μ2 and ω2 in equation (6), and the expression is as follows: Where a = 2, Δλ = 1 / 2, Δμ = π, Δφ = π / 6; j, k and i are all adjustment parameters, and (j, k, i)∈Z 4 , Z 4 is a set of integers, satisfying the following constraints:

4. The method for sparse decomposition of distribution network signals based on a combined overcomplete dictionary according to claim 3, characterized in that: In S3, performing time-frequency analysis on the Damped-sine overcomplete dictionary includes the following steps: The mathematical expression of S31 and Damped-sine atoms is: Where H(t) is a step function and satisfies the following expression: Where s3, λ3, μ3 and ω3 are the scale, time center, frequency center and phase of the Damped-sine atom respectively; S32, discretize s3, λ3, μ3 and ω3 in equation (9), and the expression is as follows: Where a = 2, Δλ = 1 / 2, Δμ = π, Δφ = π / 6; j, r, k and i are all adjustment parameters, and (j, r, k, i) ∈ Z 4 , Z 4 is a set of integers, satisfying the following constraints:

5. The method for sparse decomposition of distribution network signals based on combined overcomplete dictionary according to claim 4, characterized in that: In S4, the specific process of combining the Ex-Gabor overcomplete dictionary and the Damped-sine overcomplete dictionary for retrieval is as follows: S41. First, the overall outline of the signal to be recovered is determined by searching among the extended atoms of the Ex-Gabor overcomplete dictionary in equation (6). Then, matching is performed among the traditional Gabor atoms in equation (3) to correct the reconstruction error caused by local fluctuations. The matching process is: Where p is the number of iterations of the matching pursuit process, p ini is the upper limit of the number of iterations to search in the expanded atoms; S42. After reconstructing the periodic features and local fluctuation features of the signal based on the Ex-Gabor overcomplete dictionary, determine whether there is a mutation and oscillation in the signal; if the maximum value of the residual signal in formula (1) is less than the set value, it is determined that there is no mutation in the signal; if the maximum value of the residual signal in formula (1) is greater than or equal to the set value, it is determined that there is a mutation in the signal, and the mutation signal is reconstructed by searching for atoms in the Damped-sine overcomplete dictionary; this process is expressed as: Where, ξ th is the threshold, usually set to 20; r max is the maximum value of the residual signal R(s) in formula (1), r aver is the average value of the residual signal during the iterative process.

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