A long-period signal extraction method and system based on kernel function convolution

Abstract

The application discloses a long-period signal extraction method and system based on kernel function convolution, and belongs to the technical field of signal processing, and comprises the following steps: obtaining a time sequence containing harmonic signals and performing pretreatment; performing convolution transformation on the pretreated time sequence by using a kernel function to obtain a time-frequency matrix; calculating a time-frequency amplitude spectrum based on the time-frequency matrix, obtaining a time-frequency ridge line, and extracting a target harmonic signal from the time-frequency matrix according to the position of the time-frequency ridge line; deducting other harmonic signals similar to the period of the target harmonic signal from the time sequence containing the harmonic signals to obtain a residual time sequence, performing convolution transformation on the residual time sequence by using the kernel function, and repeating the above process to output the extracted target harmonic signal. The application effectively solves the problems of edge effect and poor extraction precision in the long-period signal extraction of the prior art, and has important scientific significance and practical application value.

Description

Technical Field

[0001] This invention belongs to the field of signal processing technology, specifically relating to a method and system for extracting long-period signals based on kernel function convolution. It is particularly suitable for the analysis of long-period signals in the field of geodesy, such as high-precision signal extraction in areas such as periodic surface deformation, changes in Earth's rotation, and core movement. Background Technology

[0002] In signal analysis within geodesy and related fields, the identification and accurate extraction of long-period signals has always been a challenging issue, especially when processing low-precision observation data. Due to limitations in early observation techniques and equipment, many weak harmonic signals may not have been accurately recorded, or the true characteristics of the signal may not have been effectively captured due to observational noise interference. Furthermore, edge effects often lead to inaccurate signal extraction during the processing of long-period signals, significantly reducing the accuracy of signal analysis. How to accurately recover the time-varying characteristics of target harmonic signals from data containing low-precision observations, particularly how to reduce edge effects in long-period signals, is a key technical challenge in signal processing.

[0003] Existing time-frequency analysis methods, such as Short-Time Fourier Transform (STFT), Wavelet Transform, and Normal Time-Frequency Transform (NTFT), are widely used in various signal analysis tasks, especially when processing short-period signals, where they can recover relatively accurate time-frequency characteristics. However, these methods fail for long-period signals. While the STFT performs well in certain frequency ranges, its fixed window size limits its frequency resolution when processing long-period signals, severely impacting accurate signal extraction. Although the Wavelet Transform offers the advantage of multi-resolution, edge effects remain significant when processing low-frequency or ultra-low-frequency signals, and its insufficient spectral distribution accuracy prevents it from effectively capturing the details of long-period signals.

[0004] Standard time-frequency transform, as a commonly used time-frequency analysis tool, can provide high-resolution time-spectrum data for signals. However, similar to wavelet transform, due to its limitations in long-period signal processing, it often struggles to effectively suppress edge effects and accurately recover the true characteristics of periodic signals. Therefore, to address these challenges, a new method is urgently needed to effectively process low-precision observation data, especially in the extraction and accurate estimation of long-period signals, providing higher accuracy and stronger stability. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of existing methods in terms of accuracy, stability, and applicability, particularly in providing ideal analysis results when dealing with long-period signals and low-precision data. This invention provides a long-period signal extraction method based on kernel function convolution. This method overcomes the limitations of traditional NTFTs in processing long-period signals by introducing a new kernel function. Through this extension, the extended standard time-frequency transform (eNTFT) not only effectively reduces spectral leakage and edge effects but also improves the extraction accuracy of time-varying features of long-period signals. Especially in the processing of low-precision data, eNTFT can more accurately recover weak harmonic signals and extract their precise frequency and amplitude characteristics.

[0006] According to one aspect of this specification, a method for extracting long-period signals based on kernel function convolution is provided, comprising:

[0007] S1. Obtain the time series containing the target harmonic signal and perform preprocessing;

[0008] S2. The preprocessed time series containing the target harmonic signal is convolved using the eNTFT kernel function to obtain the time-frequency matrix; the expression of the eNTFT kernel function is:

[0009] ,

[0010] Where t is the observation time, and t1 and t2 are the start and end times of the observation sequence, respectively. For local time, For local frequency, and For constructor, The main function is the transformation function;

[0011] S3. Calculate the time-frequency amplitude spectrum based on the time-frequency matrix and obtain the time-frequency ridge line. Extract the target harmonic signal from the time-frequency matrix according to the position of the time-frequency ridge line.

[0012] Furthermore, the method further includes S4:

[0013] Subtract other harmonic signals whose periods are within the set time error range from the time series containing the target harmonic signal to obtain the residual time series. Repeat steps S2-S3 to output the extracted target harmonic signal.

[0014] Furthermore, the expression for the time-frequency matrix is:

[0015]

[0016] The horizontal line indicates the complex conjugate operation, and * indicates convolution. For observation time, For local time, For local frequency, Represents the observed time series, Represents a local time sequence. Represents the eNTFT kernel function. and For constructor, This is the main function for transformation.

[0017] Furthermore, the aforementioned and Replace with and The expression is:

[0018] ,

[0019]

[0020] in, For observation time, and These are the start and end times of the observation sequence, respectively. For local frequency, It is an imaginary number.

[0021] Further, the time-frequency amplitude spectrum is calculated based on the time-frequency matrix, and the time-frequency ridge is obtained. The target harmonic signal is extracted from the time-frequency matrix according to the position of the time-frequency ridge, including:

[0022] Calculation of time-frequency amplitude spectrum based on time-frequency matrix;

[0023] The number of target harmonic signals is determined by the number of time-frequency ridges in the time-frequency amplitude spectrum;

[0024] The period, occurrence, and end time of the target harmonic signal are determined based on the location of the time-frequency ridge.

[0025] The target harmonic signals are extracted from the time-frequency matrix based on the number, period, and occurrence and termination times of the target harmonic signals.

[0026] Furthermore, the expression for the time-frequency amplitude spectrum is:

[0027]

[0028] in, The time-frequency amplitude spectrum, Let be the time-frequency matrix after convolution with the kernel function, where || represents the absolute value of each element in the matrix.

[0029] Furthermore, extracting the target harmonic signal from the time-frequency matrix also includes: if the time series containing the target harmonic signal is a real number sequence, performing a real part extraction operation on the extracted target harmonic signal and multiplying it by 2.

[0030] Furthermore, the time-frequency ridges are several ridges with maximum values ​​formed by taking extreme values ​​of the amplitude spectrum along the horizontal axis direction.

[0031] According to one aspect of this specification, a system for extracting long-period signals based on kernel function convolution is provided, comprising:

[0032] The signal preprocessing module is used to acquire and preprocess the time series containing the target harmonic signal;

[0033] The convolution transform module is used to perform a convolution transform on the preprocessed time series containing the target harmonic signal using the eNTFT kernel function to obtain a time-frequency matrix; the expression of the eNTFT kernel function is:

[0034] ,

[0035] Where t is the observation time, and t1 and t2 are the start and end times of the observation sequence, respectively. For local time, For local frequency, and For constructor, The main function is the transformation function;

[0036] The long-period signal extraction module is used to calculate the time-frequency amplitude spectrum based on the time-frequency matrix and obtain the time-frequency ridge line, and extract the target harmonic signal from the time-frequency matrix according to the position of the time-frequency ridge line.

[0037] According to one aspect of this specification, an electronic device is provided, including a memory and a processor, the memory storing a computer program, the processor executing the computer program to implement the steps of the long-period signal extraction method based on kernel function convolution.

[0038] Compared with the prior art, the beneficial effects of the present invention are:

[0039] 1. The eNTFT kernel function proposed in this invention significantly reduces the strong edge effects that traditional time-frequency analysis methods, such as STFT, wavelet transform, and NTFT, tend to produce when processing long-period signals, and extracts the periodic features of the signal more accurately. Compared with other complex nonlinear time-frequency analysis methods, the eNTFT kernel function convolution method is computationally simple, efficient, and can process large-scale data in a shorter time, making it more adaptable and suitable for applications in fields such as geodesy and geophysical signals, earthquake research, and satellite navigation.

[0040] 2. This invention combines eNTFT kernel function convolution with time-frequency analysis to achieve high-precision extraction of harmonic signals, especially for long-period signals, accurately recovering the time-varying characteristics of the target harmonic signal. Furthermore, it is easy to program and implement, with a clear algorithm structure, low hardware requirements, and rapid deployment on common computing platforms, making it suitable for various signal processing applications. It is not only applicable to the extraction of long-period signals in geodesy, such as the analysis of geophysical signals like day length and polar motion, but can also be widely applied to time-series data analysis in other scientific fields, such as astrophysics and meteorology, demonstrating broad application prospects.

[0041] 3. This invention further improves the accuracy of the target harmonic signal by subtracting interference signals from the original data and extracting them repeatedly, ensuring the accuracy and reliability of the results. This iterative optimization process can significantly improve the extraction effect of long-period signals in practical applications, and has important scientific significance and practical application value. Attached Figure Description

[0042] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0043] Figure 1 This is a flowchart illustrating a signal extraction method based on kernel function convolution, provided in an embodiment of the present invention.

[0044] Figure 2 A synthesized time series diagram is provided for an embodiment of the present invention.

[0045] Figure 3 The time-frequency amplitude spectrum of the synthesized sequence provided in the embodiment of the present invention.

[0046] Figure 4 The diagram shows the input signal provided in the embodiments of the present invention and the corresponding signal extracted from the synthesized time series using NTFT and eNTFT, respectively. Detailed Implementation

[0047] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0048] like Figure 1As shown, this embodiment of the invention provides a method for extracting long-period signals based on kernel function convolution, including: Signal preprocessing: Preprocessing the input 1D time series containing harmonic signals to ensure data quality. These time series can come from geodetic observation data, such as diurnal variation, polar motion data, sea level change, etc. The preprocessing steps include denoising, smoothing, and other necessary signal correction operations to prepare for subsequent convolution operations. Convolution transformation: Convolving the preprocessed time series with a kernel function to obtain a convolved time-frequency matrix. Through kernel function convolution, the time-frequency features of the signal can be effectively decomposed, and edge effects in the signal can be significantly suppressed, thereby improving the extraction accuracy of long-period signals. Time-frequency domain analysis: Plotting an amplitude spectrum in the convolved time-frequency matrix and analyzing the ridges in the spectrum. Based on the position and characteristics of the ridges, determining the number, period, and occurrence and end times of the harmonic signals contained in the signal. Signal extraction: Extracting the target harmonic signal directly from the convolution transformation matrix based on the position corresponding to the ridge. This method enables accurate extraction of frequency and amplitude information from time-varying signals, especially under low-precision data conditions, effectively recovering weak periodic signals. Signal Accuracy Improvement: To further improve the extraction accuracy of the target harmonic signal, after extraction, other harmonic signals with periods similar to the target harmonic signal (within a set time error range) are subtracted from the original data to avoid overlapping interference. This process is repeated to accurately extract the characteristics of the target harmonic signal. Output Results: Finally, the precise characteristics of the extracted target harmonic signal, including parameters such as period, phase, and amplitude, are output, providing high-precision data support for further signal analysis or geophysical research.

[0049] Specifically, embodiments of the present invention provide a step after inputting a signal: inputting a time series containing several harmonic signals. This signal can be a geophysical signal, such as a seismic signal, tidal signal, etc. The signal can be represented in the following form:

[0050] (1)

[0051] In the formula, This represents an observation time series containing M harmonic signals, h k It is the kth harmonic. , and These represent the complex amplitude, rounded frequency, and phase of the harmonic, respectively, while e(t) is the noise. This signal format encompasses typical harmonic signals from various geophysical phenomena, making it suitable for a wide range of scientific and engineering applications.

[0052] Specifically, embodiments of the present invention provide a convolution transformation for the input signal, with the following specific steps:

[0053] Convolution is implemented based on the Fast Fourier Transform (FFT), and the kernel function is an improved technique that can effectively mitigate the effect of the rapid decrease in extraction accuracy at both ends of the observed sequence. The formula is:

[0054] (2)

[0055] In the formula, For observation time, and These are the start and end times of the observation sequence, respectively. For local time, For local frequency, and These are all corresponding constructors, used replace The specific form is as follows:

[0056] (3)

[0057] (4)

[0058] Transformation main function for:

[0059] (5)

[0060] In the formula, σ is an imaginary number, and σ is the window width factor.

[0061] Based on the Fast Fourier Transform (FFT), the input signal is convolved with a kernel function to convert it into a time-frequency domain signal. The convolutional form (time-frequency matrix) of the eNTFT is defined as follows:

[0062] (6)

[0063] In the formula, the upper horizontal line represents the complex conjugate operation, and * represents convolution; It is a kernel function, used in the formula. Replace t in formula (2) as a whole; and For the constructors in formulas (3) and (4), The transform master function in formula (5) is τ, which replaces t. The eNTFT kernel function differs from the NTFT kernel function. For time-frequency resolution, the latter depends only on the local frequency, while the former can vary with both local time and frequency, making its kernel function more flexible. The Fourier transform of the eNTFT kernel function is defined as:

[0064] (7)

[0065] In the formula, ω represents the frequency of the Fourier transform, which satisfies the following property:

[0066] (8)

[0067] (9)

[0068] Formula (8) means that the Fourier transform frequency ω and the local frequency are equal if and only if ... When they are the same, the Fourier transform of the kernel function =1; Formula (9) means that when the Fourier transform frequency ω and the local frequency are 1 Different absolute values Less than 1. Based on the properties of kernel functions, it is found that after the observed signal undergoes a convolution transformation, when the local frequency and the frequency of the k-th harmonic are the same, the following will be obtained:

[0069] (10)

[0070] Therefore, we can obtain the following two conditions:

[0071] (11)

[0072] (12)

[0073] Formula (11) states that the absolute value of the time-frequency matrix reaches a maximum value and is the same as the amplitude of the harmonic if and only if the local frequency is the same as the frequency of the k-th harmonic. Formula (12) indicates that when the local frequency is the same as the frequency of the kth harmonic, it can be calculated along the local time. Obtain the kth harmonic from the time-frequency matrix.

[0074] Specifically, embodiments of the present invention provide spectral analysis and ridge line acquisition. In the time-frequency domain, the ridge line distribution is obtained by analyzing the amplitude spectrum. The specific steps are as follows:

[0075] The formula for calculating the amplitude spectrum is as follows:

[0076] (13)

[0077] in, It is the time-frequency matrix after convolution with the kernel function, where || represents the absolute value of each element in the matrix. This spectrum reflects the combined amplitude distribution of the signal at different times and frequencies, and is a key indicator in time-frequency analysis. According to formulas (8)-(9) and (11)-(12), it can be found that when there is a harmonic signal h kWhen the frequency changes, a line resembling a ridge will form along the τ direction, which is called the ridge line of the time-frequency transformation. The ordinate corresponding to this ridge line is the harmonic signal h. k frequency β k .

[0078] Specifically, the embodiments of the present invention provide the following details regarding parameter estimation:

[0079] The ridge obtained by taking the maximum value will also have a frequency that changes with time, that is, the time-varying frequency β is obtained. k (τ), from which the time-varying period T is estimated. k (τ)=2π / β k (τ). Based on this, the complex form h of the harmonics can be obtained from the convolved time-frequency matrix. k Even if the input is a real harmonic, the phase φ of the signal can be estimated by taking the angle of the complex harmonic. k (τ)=Angle(h k (τ)) estimates the amplitude envelope of the signal by taking its absolute value.

[0080] Specifically, the embodiments of the present invention provide the specific content of signal extraction:

[0081] In fact, based on the kernel function used, its Fourier transform usually also satisfies the following properties:

[0082] (14)

[0083] That is, when ω distance When the distance is relatively far, the Fourier transform of the kernel function is generally 0. Substituting this into formula (10) yields:

[0084] (15)

[0085] This means that the ridge of the eNTFT spectrum along the k-th harmonic is obtained from the convolved time-frequency matrix. It is precisely the k-th harmonic itself. When the input time series is a real number sequence,

[0086] (16)

[0087] Formula (16) reveals that at the k-th ridge (corresponding to the vertical axis) The real number sequence obtained from the kernel function is exactly half of the actual real harmonic amplitude, so to restore the true amplitude, it needs to be doubled. When the input signal is a complex sequence, signal extraction is performed directly. Since the kernel function convolution process is implemented through a sliding window, that is, in time-segmented manner, the time-varying characteristics of the harmonic signal, including amplitude, period, and phase, can be analyzed.

[0088] Specifically, the embodiments of the present invention provide the following details regarding repeated extraction:

[0089] Since formula (14) is only an approximation, the Fourier transform of the kernel function cannot be strictly zero at the two edges of the signal. This means that there are also errors in the harmonics extracted at the signal edges. Through the above steps, the preliminary extraction results of all signals can be obtained. For a certain signal h to be studied... l In this regard, accurate valuations can be obtained through repeated extraction. The specific steps are as follows:

[0090] Obtain the target harmonic signal h l (t) Preliminary extraction results of several nearby signals with similar periods h l-1 (t) and h l+1 (t) etc., these signals are subtracted from the original observations to obtain the residual time series. …This is equivalent to subtracting most of the stable interference signals from the original data, even at the signal edges. While not strictly zero, its value is close to zero, which effectively suppresses residual errors after subtracting interference signals. Repeated extraction allows for precise extraction of the target harmonic signal and accurate estimation of its time-varying characteristics.

[0091] Specifically, embodiments of the present invention output estimated time-varying amplitude, period, phase, and the extracted complete signal. These parameters are key results of signal analysis and can provide important basis for subsequent scientific research or engineering applications. The accuracy evaluation of the estimation results includes calculating the deviation between the extracted signal and the theoretical signal value.

[0092] Specifically, the embodiments of the present invention were verified by synthesizing time series of actual signals showing noise and diurnal variations. The specific steps are as follows:

[0093] First, four periodic signals (15.8 years, 18.6 years, 22.3 years, and 33 years), long-term linear trends, and random white noise (average amplitude in the frequency domain of 0.02 ms) were synthesized, spanning from May 1880 to May 2019, with a sampling interval of one year. The periodic signal s i (t)=A i (t)cos(2π / P i t+φ i The simulation parameters for (i=1,2,3,4) are shown in Table 1. These parameters were identified from ΔLOD. The 18.6-year period is caused by lunar tides and is simulated as a stable standard harmonic based on its physical properties. The remaining signals are simulated as quasi-harmonics with time-varying amplitudes. The 22.3-year periodic signal has a decaying amplitude. The synthesized results are shown in Table 1. Figure 2 As shown.

[0094] In order to reconstruct the harmonic information without knowing the signal information contained in the simulated time series, it is first necessary to determine which periodic information is contained in the simulated time series. A convolution transformation can be used to obtain the transformed time-frequency matrix, and the amplitude spectrum can be obtained by taking its absolute value. Figure 3 As shown, the bright bars represent the ridges of the synthesized signal after eNTFT transformation. Four prominent ridges can be seen, with corresponding periods of 15.8 years, 18.6 years, 22.3 years, and 33 years, which are consistent with the input period. This verifies the ability of the eNTFT spectrum to detect harmonic signals.

[0095] The input periodic signal was recovered from the synthesized time series using NTFT and eNTFT respectively, and the results are as follows. Figure 4 As shown in the figure, the signal recovered using eNTFT almost overlaps with the corresponding input signal, while the result recovered using NTFT deviates somewhat from the input. Furthermore, the figure also verifies that the signal recovered using NTFT may experience phase shifts within the edge effect region, while eNTFT can accurately estimate the time-varying amplitude and phase of the signal.

[0096] Specifically, the method of this invention is applicable to geodesy and geophysical signal analysis, such as Earth's rotation, GPS surface deformation, and sea-level change. For example, in Earth's rotation, this method can be used to analyze long-period changes in Earth's rotation caused by core movement, thereby inferring the strength of the magnetic field in the core; in GPS surface deformation, this method can be used to analyze global long-period deformation and spatial patterns, thereby establishing a global deformation field model and improving GPS accuracy; in sea-level change research, this method can extract periodic changes in sea level caused by climate change. Furthermore, this method can also be applied to research fields such as communication engineering and biomedicine, possessing broad application prospects and significant scientific value.

[0097] Specifically, this invention utilizes kernel function convolution to refine the time-frequency distribution of signals, enabling clear identification of the start time and amplitude characteristics of signals with different periods. It exhibits a significant advantage in extracting weak harmonic signals, particularly in early, low-precision data. Simulation experiments verified the reliability of this method in signal records across different time periods, demonstrating its ability to recover true periodic signal information under complex data conditions. The eNTFT method is suitable for geophysical data analysis, such as studies of Earth's core movement, and can also be extended to other time-series signal analysis fields. This method is simple, efficient, and has strong practical application potential, especially in improving signal extraction accuracy in low-precision observation data processing.

[0098] The implementation of the various embodiments of the present invention is based on programmed processing by a device with processor functionality. Therefore, in practical engineering, the technical solutions and functions of the various embodiments of the present invention are encapsulated into various modules. Based on this reality, and building upon the above embodiments, the embodiments of the present invention provide a long-period signal extraction method system based on kernel function convolution. This system is used to execute a long-period signal extraction method based on kernel function convolution from the above method embodiments.

[0099] The system includes, according to one aspect of this specification, a long-period signal extraction method system based on kernel function convolution, comprising: a signal preprocessing module for acquiring and preprocessing a time series containing a target harmonic signal; a convolution transformation module for performing a convolution transformation on the preprocessed time series containing the target harmonic signal using an eNTFT kernel function to obtain a time-frequency matrix; and a signal extraction module for calculating the time-frequency amplitude spectrum based on the time-frequency matrix and obtaining time-frequency ridges, and extracting the target harmonic signal from the time-frequency matrix according to the position of the time-frequency ridges.

[0100] The long-period signal extraction method system based on kernel function convolution provided in this invention addresses the shortcomings of existing methods in terms of accuracy, stability, and applicability, particularly in providing ideal analysis results when dealing with long-period signals and low-precision data. This system employs several modules, combining kernel function convolution with time-frequency analysis, to extract harmonic signals with high precision. Especially for long-period signals, it can accurately recover the time-varying characteristics of the target harmonic signal. By subtracting interference signals from the original data and repeating the extraction process, the accuracy of the target harmonic signal can be further improved, ensuring the accuracy and reliability of the results. This iterative optimization process can significantly improve signal extraction performance in practical applications.

[0101] Based on the same inventive concept as the foregoing embodiments, this embodiment of the invention also provides an electronic device, including a memory and a processor. The memory is used to store computer-executable instructions, and the processor is used to execute the computer-executable instructions to realize a long-period signal extraction method based on kernel function convolution as proposed in the above embodiments.

[0102] This invention also provides a computer-readable storage medium storing a computer program thereon. When executed by a processor, this program effectively solves the problems of edge effects and poor extraction accuracy in the prior art for long-period signal extraction, and has significant scientific and practical application value.

[0103] Finally, it should be noted that the above specific embodiments are merely representative examples of the present invention. Obviously, the present invention is not limited to the above specific embodiments and many variations are possible. Any simple modifications, equivalent changes, and alterations made to the above specific embodiments based on the technical essence of the present invention should be considered within the protection scope of the present invention.

Claims

1. A method for extracting long-period signals based on kernel function convolution, characterized in that, include: S1. Obtain the time series containing the target harmonic signal and perform preprocessing; The time series is derived from geodetic observation data, including sea level changes; the time series includes multiple harmonic signals, including tidal signals; S2. The preprocessed time series containing the target harmonic signal is convolved using the eNTFT kernel function to obtain the time-frequency matrix; the expression of the eNTFT kernel function is: ， Where t is the observation time, and t1 and t2 are the start and end times of the observation sequence, respectively. For local time, For local frequency, and For constructor, The transformation main function; where, using Replacement constructor and In The specific form is as follows: ， ， Transformation main function The expression is: ， In the formula, σ is an imaginary number, and σ is the window width factor; S3. Calculate the time-frequency amplitude spectrum based on the time-frequency matrix and obtain the time-frequency ridge line. Extract the target harmonic signal from the time-frequency matrix according to the position of the time-frequency ridge line.

2. The long-period signal extraction method based on kernel function convolution according to claim 1, characterized in that, The method further includes S4: Subtract other harmonic signals whose periods are within the set time error range from the time series containing the target harmonic signal to obtain the residual time series. Repeat steps S2-S3 to output the extracted target harmonic signal.

3. The method for extracting long-period signals based on kernel function convolution according to claim 1, characterized in that, The expression for the time-frequency matrix is: ， The horizontal line indicates the complex conjugate operation, and * indicates convolution. For observation time, For local time, For local frequency, Represents the observed time series, Represents a local time sequence. Represents the eNTFT kernel function. and For constructor, This is the main function for transformation.

4. The method for extracting long-period signals based on kernel function convolution according to claim 1, characterized in that, The time-frequency amplitude spectrum is calculated based on the time-frequency matrix, and the time-frequency ridge is obtained. The target harmonic signal is extracted from the time-frequency matrix according to the position of the time-frequency ridge, including: Calculation of time-frequency amplitude spectrum based on time-frequency matrix; The number of target harmonic signals is determined by the number of time-frequency ridges in the time-frequency amplitude spectrum; The period, occurrence, and end time of the target harmonic signal are determined based on the location of the time-frequency ridge. The target harmonic signals are extracted from the time-frequency matrix based on the number, period, and occurrence and termination times of the target harmonic signals.

5. The long-period signal extraction method based on kernel function convolution according to claim 4, characterized in that, The expression for the time-frequency amplitude spectrum is: ， in, The time-frequency amplitude spectrum, Let be the time-frequency matrix after convolution with the kernel function, where || represents the absolute value of each element in the matrix.

6. The long-period signal extraction method based on kernel function convolution according to claim 4, characterized in that, Extracting the target harmonic signal from the time-frequency matrix also includes: if the time series containing the target harmonic signal is a real number sequence, performing a real part extraction operation on the extracted target harmonic signal and multiplying it by 2.

7. The method for extracting long-period signals based on kernel function convolution according to claim 1, characterized in that, The time-frequency ridges are several ridges with maximum values ​​formed by taking extreme values ​​of the amplitude spectrum along the horizontal axis.

8. A long-period signal extraction system based on kernel function convolution, characterized in that, The method for extracting long-period signals based on kernel function convolution as described in any one of claims 1 to 7 includes: The signal preprocessing module is used to acquire and preprocess the time series containing the target harmonic signal; The convolution transform module is used to perform a convolution transform on the preprocessed time series containing the target harmonic signal using the eNTFT kernel function to obtain a time-frequency matrix; the expression of the eNTFT kernel function is: ， Where t is the observation time, and t1 and t2 are the start and end times of the observation sequence, respectively. ϖ is the local time, where ϖ is the local frequency. and For constructor, The main function is the transformation function; The long-period signal extraction module is used to calculate the time-frequency amplitude spectrum based on the time-frequency matrix and obtain the time-frequency ridge line, and extract the target harmonic signal from the time-frequency matrix according to the position of the time-frequency ridge line.

9. An electronic device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the steps of the long-period signal extraction method based on kernel function convolution as described in any one of claims 1 to 7.

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