Frequency modulation wavelet transform time-frequency analysis method based on local entropy selection scale extraction

Abstract

The invention discloses a frequency modulation wavelet transform time-frequency analysis method based on local entropy selection scale extraction, belongs to the technical field of signal processing, and particularly relates to the problem of insufficient time-frequency representation precision of non-stationary signals. According to the method, a sliding Gaussian window function matrix is constructed, a frequency modulation angle is discretized to generate a telescopic primary function, and a multi-angle time-frequency sub-representation matrix is generated in combination with frequency modulation wavelet transform; after Fourier transform, constructing a redistribution operator by using a partial derivative and a Dirichlet function, and extracting time-frequency points with strong energy and stable phases to form a high-resolution time-frequency distribution matrix; and finally, time-frequency blocks are divided, Renyi entropy is calculated, and optimal time-frequency blocks with the minimum entropy value are selected to be spliced into a complete time-frequency representation matrix. The method is mainly used for improving the time-frequency resolution and energy concentration of non-stationary signals (such as mechanical fault diagnosis and biomedical signals).

Description

Technical Field

[0001] This invention relates to the field of signal processing technology. More specifically, this invention relates to a time-frequency analysis method based on frequency-modulated wavelet transform extracted using local entropy-selective scale extraction. Background Technology

[0002] Non-stationary signal analysis in fields such as rotating machinery, radar, and biomedical engineering has always been a hot topic in academia and industry. The dynamic change in frequency over time of non-stationary signals makes traditional Fourier transforms significantly limited in processing such signals. Therefore, time-frequency analysis techniques are widely used to map one-dimensional signals into two-dimensional time and frequency spaces, using combined time and frequency information to describe signal characteristics, thus enabling the analysis of time-varying non-stationary vibration signals. However, traditional time-frequency analysis methods (such as short-time Fourier transform, wavelet transform, and Wigner-Ville distribution) are often limited by the Heisenberg uncertainty principle, resulting in blurred energy distribution in the time-frequency plot, making it difficult to accurately characterize the instantaneous frequency characteristics of the signal.

[0003] To address the aforementioned issues, the Chirplet Transform (CT) improves upon traditional basis functions by introducing a frequency modulation parameter, enabling it to better adapt to signals with linear frequency modulation characteristics. However, CT faces challenges when processing multi-component signals, as its fixed basis function form limits its ability to describe complex frequency modulation trajectories. Based on this, the Scaling-Basis Chirplet Transform (M. Li, T. Wang, F. Chu, Q. Han, Z. Qin and MJ Zuo, "Scaling-Basis Chirplet Transform," in IEEE Transactions on Industrial Electronics, vol. 68, no. 9, pp. 8777-8788, Sept. 2021.) was proposed, extending the basis function design of CT and allowing the basis functions to more accurately match the instantaneous frequency trajectory of the signal through adjustments to time and frequency scales. Scaling-Basis Chirplet Transform (SBCT) significantly improves the energy concentration of time-frequency representation, exhibiting excellent time-frequency resolution even when dealing with signals with strong background noise. However, even SBCT has certain limitations in capturing the signal's modulation characteristics. To further improve the accuracy of time-frequency representation, EMCT (Ya He, Minghui Hu, Zhinong Jiang, Kun Feng, Xuan Ming, Local maximum synchronization queezes from entropy matching chirplet transform, Mechanical Systems and Signal Processing, Volume 181, 2022, 109476, ISSN 0888-3270.) optimizes the basis function by minimizing the Rényi entropy, making its energy more concentrated on the actual instantaneous frequency trajectory. Therefore, EMCT exhibits extremely high accuracy in selecting the signal modulation frequency. Although EMCT has made significant progress in optimizing energy concentration, ambiguity components in the time-frequency plot still affect the further extraction of signal features.Therefore, SET (G. Yu, M. Yu and C. Xu, "SynchroextractingTransform," in IEEE Transactions on Industrial Electronics, vol. 64, no. 10, pp. 8042-8054, Oct. 2017.) is used as a post-processing method to further optimize the results of EMCT. SET achieves higher energy concentration and a more accurate representation of the instantaneous frequency trajectory by redistributing energy to remove the diffusion component in the time-frequency plot.

[0004] Traditional methods such as short-time Fourier transform, wavelet transform, and Wigner-Ville distribution are constrained by the Heisenberg uncertainty principle, making it difficult to balance time and frequency resolution. This leads to the diffusion of time-frequency energy distribution, failing to accurately characterize the instantaneous frequency trajectory of the signal. Linear frequency-modulated wavelet transform improves basis function design by introducing frequency modulation parameters, but its fixed-form basis functions struggle to match complex nonlinear frequency modulation patterns, especially with multi-component signals. While global entropy optimization-based methods (such as EMCT) improve overall energy concentration, residual ambiguity components still interfere with accurate instantaneous frequency extraction because they do not consider local time-frequency characteristics. Set-to-sequence (SET) as a post-processing technique can sharpen the time-frequency trajectory, but its effectiveness depends on the quality of the preceding time-frequency representation. When local energy diffusion exists in the input, SET struggles to completely eliminate interference components and cannot address the fundamental problem of local feature mismatch in global optimization. The nonlinear frequency modulation characteristics of the signal and dynamic noise environments further exacerbate local energy diffusion, limiting the improvement of time-frequency resolution. Summary of the Invention

[0005] The purpose of this invention is to provide a time-frequency analysis method for frequency-modulated wavelet transform based on local entropy-selective scale extraction. This method combines the flexibility of scale-based wavelet transform, the high energy concentration of entropy matching method, and the post-processing capability of synchronous extraction transform. By optimizing the basis function design, entropy selection, and post-processing strategies, the time-frequency resolution and accuracy of non-stationary signals are improved.

[0006] This solves the problem that traditional methods cannot match all instantaneous frequency components when analyzing nonlinear multi-fundamental frequency signals.

[0007] This addresses the issues of ambiguity and energy divergence that arise when traditional methods analyze nonlinear frequency modulated signals, particularly when the instantaneous frequency components are closely spaced.

[0008] This addresses the problems of energy dispersion and insufficient time-frequency resolution in traditional time-frequency analysis methods for non-stationary signals.

[0009] Solve the problem of time-frequency analysis distortion caused by signal boundary truncation.

[0010] This addresses the problem of insufficient matching of nonlinear frequency modulation signals caused by a fixed basis function angle.

[0011] This addresses the issue of decreased accuracy caused by energy leakage and phase abrupt changes in time-frequency representation.

[0012] This addresses the problem that global entropy optimization cannot adapt to local time-frequency characteristics.

[0013] Solve the problem of boundary discontinuity caused by direct splicing of optimal time-frequency blocks.

[0014] Solve the energy jump problem at the junction of adjacent time-frequency blocks.

[0015] To address the aforementioned problems and achieve the objectives and other advantages of this invention, a time-frequency analysis method based on frequency-modulated wavelet transform extracted using local entropy selection is provided, comprising the following steps: Step 1) Obtain the sampling data of the target signal and generate a signal matrix according to the set sampling frequency and sampling time; Step 2) Generate a sliding Gaussian window function, and apply the sliding Gaussian window function to the signal matrix to generate a sliding window function matrix; Step 3) Discretize the frequency modulation angle range into multiple angle intervals, construct scaling basis functions for different angles based on frequency modulation wavelet transform, and match the scaling basis functions with the sliding window function matrix point by point to obtain the time-frequency sub-representation matrix for multiple angles; Step 4) Perform a Fourier transform on the time-frequency sub-representation matrix to generate a complex time-frequency representation matrix; Step 5) Calculate the partial derivatives of the time-frequency representation matrix and construct a redistribution operator based on the partial derivatives. Then, introduce the Dirichlet function to redefine the redistribution operator, re-express the partial derivatives, and constrain the phase derivatives. Then, extract time-frequency points with strong energy and stable phase changes from the original time-frequency matrix, and after dynamic adjustment, construct a time-frequency distribution matrix with good energy concentration characteristics. Step 6) Divide the time-frequency distribution matrix into several time-frequency blocks, calculate the Rényi entropy of each time-frequency block, and select the time-frequency block generated by the basis function angle with the smallest entropy value as the optimal time-frequency block; Step 7) Segment all the optimal time-frequency blocks to form a complete high-resolution time-frequency representation matrix.

[0016] Preferably, in the time-frequency analysis method of frequency-modulated wavelet transform based on local entropy selection scale extraction, in step 1), a signal matrix is ​​generated according to the set sampling frequency and sampling time, and zero-padding is performed at both ends of the signal to generate an extended signal; In step 2), a sliding Gaussian window function is generated, and the extended signal is processed by the sliding Gaussian window function to generate a sliding window function matrix; The specific steps include: a. Based on the set sampling frequency f s and sampling time t tend Acquire target signal S ( t ), generating the original time-domain signal matrix; b. Zero-padded both ends of the original time-domain signal matrix, with a padding length of... N w / 2, generating extended signal S est ( t The expression is: S est ( t ) = [ s 1, s 2,…, s M ] in, M = f s ﹒t tend , where is the number of sampling points for the signal. N w The length of the Gaussian window function. s M For the original signal at the 1st M The values ​​of each sampling point; c. Select a length of N w The Gaussian window function is used, and the overlap length of the Gaussian window function is set to... N ov Calculate the step size of the Gaussian window function. K 1 = N w - N ov Next, calculate the number of windows that can be slidable. K = ( (N - K 1 ) / K 1), of which K Rounding down, the sliding Gaussian window function is expressed as: Where σ is the window function width factor, N Original signal S ( t ) length; d. Using a sliding Gaussian window function, starting from the beginning of the extended signal, with a step size of... K 1. Slide the signal to cover the entire extended signal. Each slide extracts a local segment of the extended signal, multiplies it by the window function to obtain a weighted signal, and after traversal, generates a sliding window function matrix, expressed as: in, g ( t - t c ) as t c A sliding Gaussian window function centered at the center. t c Centered on time.

[0017] Preferably, in the time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform, in step 3), the frequency modulation angle range [-π / 2, π / 2] is discretized into... N na Each angle interval is divided into several angle sequences, and each discrete angle is... η m The calculation formula is: Based on frequency-modulated wavelet transform, scaling basis functions at different angles are constructed. The expressions for the scaling basis functions are as follows: Among them, tan ( η m To adjust the amplitude of nonlinear frequency modulation, f For frequency, i The imaginary unit; By point-by-point matching of the scaling basis functions with the sliding window function matrix, a multi-angle time-frequency sub-representation matrix is ​​obtained. The expression for the time-frequency sub-representation matrix is ​​as follows: .

[0018] Preferably, in the time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform, in step 4), the time-frequency representation matrix expression is: in, j The imaginary part of a complex number ， These are basis functions.

[0019] Preferably, in the time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform, in step 5), the partial derivatives of the time-frequency representation matrix obtained in step 4) are calculated. The expression is: The redistribution operator is constructed based on partial derivatives, and its expression is: The redistribution operator is redefined using the Dirichlet function in SET, with the expression: Therefore, partial derivatives To further express: And it satisfies the following constraints: The effective time-frequency points with strong energy and stable phase changes are extracted from the original time-frequency representation matrix, expressed as follows: in, Delta For discrete frequency intervals, For a given time center t c and frequency center f c Below is the instantaneous frequency estimate obtained through the SET method; After dynamic adjustment, a time-frequency distribution matrix with energy concentration characteristics is constructed, expressed as: .

[0020] Preferably, in the time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform, in step 6), based on the time-frequency distribution matrix... number of rows k The time-frequency distribution matrix is ​​divided into several time-frequency blocks, and the number of blocks in the vertical direction is determined. N nd Number of horizontal blocks N nc And calculate the number of rows in each time-frequency block. N h Number of columns N hh The calculation formula is: , If the number of rows in the time-frequency distribution matrix k If it is divisible by the number of blocks, use the result of the divisibility; otherwise, round down. from T e Extract the block matrix and evaluate its entropy value. T eTo obtain the three-dimensional matrix composed of the effective time-frequency point amplitudes under amplitude threshold and phase derivative constraints, the Rényi entropy formula is used: in, α Let be the order of Rényi entropy, with a value of 3. P i The energy distribution probability of the time-frequency module is calculated using the following formula: R e To further express: Next, each time-frequency block is traversed, the corresponding Rényi entropy value is calculated, and the time-frequency block generated by the basis function angle with the smallest entropy value is selected as the optimal time-frequency block.

[0021] Preferably, in the time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform, in step 7), all optimal time-frequency blocks are concatenated to form a complete high-resolution time-frequency representation matrix, the expression of which is: .

[0022] Preferably, in the time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform, after step 7), the method further includes: Step 8) Reassignment operator based on Dirichlet function The complete high-resolution time-frequency representation matrix TFPR opt ( f c , t c The data is redistributed into a new time-frequency representation matrix, expressed as: . Preferably, in the time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform, step 7) includes the following steps in the splicing process: A. Boundary parameter setting: Define the boundary time points of adjacent time-frequency blocks t b = m × N h × △ t ; in, m The time-frequency block number, 1 ≤ m≤N nd -1, △ t For time resolution, △t = 1 / f s ; Calculate the number of overlapping points ; Calculate the overlap time half-width; ; B. Window function construction: In the global time coordinate t interval [ t b -Δ T , t b + Δ T Within, according to time resolution △ t Values ​​are taken at equal intervals; Construct the left window function: Construct the right window function: C. Fusion computing: For global time coordinates t and frequency index p Perform fusion computing: Where, 1 ≤ p ≤ N f , , B m ( t , p ) is the first m The optimal time-frequency block is at point ( t , p The value at ) B m+1 ( t , p ) is the first m+ One optimal time-frequency block at point ( t , p The value at ); D. Matrix construction: Generate the final time-frequency matrix: .

[0023] Preferably, in the time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform, after step 6) and before step 7), a phase continuity calibration step is further included, which includes: S1. Determine the boundary between adjacent blocks: For two adjacent optimal time-frequency blocks generated in step 6), identify their natural intersection time point as the boundary time point. t b ; S2. Calculate the boundary frequency derivative: Calculate the previous optimal time-frequency block TFPR prev ( f , t At the boundary time point t b Right derivative of instantaneous frequency at point: ωL Calculate the next optimal time-frequency block TFPR next ( f , t At the boundary time point t b Left derivative ωR of instantaneous frequency at point: S3. Dynamic phase compensation: If the absolute value of the derivative difference |ωL-ωR| > preset tolerance threshold ε Then, a complex domain phase rotation operation is performed on the next optimal time-frequency block to generate a compensated time-frequency block. TFPR comp Its expression is: in, t For time variables, f For frequency variables, l It is the imaginary unit and satisfies l 2 = -1; S4. Replace the compensation block: Replace the compensated time-frequency block TFPR comp ( t , f This is used as the new next optimal time-frequency block.

[0024] The present invention has at least the following beneficial effects: This invention innovatively introduces a local entropy selection strategy and a multi-angle basis function matching mechanism, significantly optimizing the dynamic adaptation capability between basis functions and signal characteristics. It breaks through the traditional time-frequency resolution limitations when processing multi-component signals and signals with strong modulation characteristics. By employing a block entropy optimization strategy to achieve precise matching between local signal characteristics and basis functions, it effectively solves international challenges such as multi-component aliasing in rotating machinery composite faults and weak feature extraction in biomedical applications, providing a novel solution for radar multi-target resolution.

[0025] This invention creatively constructs an adaptive basis function matrix based on entropy minimization to achieve millisecond-level dynamic tracking of instantaneous frequency characteristics. By generating tunable scaling basis functions in real time, it completely eliminates fuzzy component interference and noise propagation in complex scenarios such as wind power variable speed bearings, and overcomes the representation mismatch problem of modulation signals in the region of sudden speed changes, significantly outperforming the limited architecture of existing linear frequency modulated wavelet transforms.

[0026] This invention deeply integrates the sharpening capabilities of Synchronous Extraction Transform (SET) with the flexibility of Scale-Based Frequency Transform (SBCT), forming a practical technical system that combines high resolution and strong reconstructibility. Its unique triple-protection mechanism—gradient window boundary fusion, dynamic phase calibration, and amplitude / phase dual constraints—enables precise separation of multi-component signals such as gearbox composite faults, maintaining excellent robustness even in high-noise environments. Through a zero-fill strategy and basis function pre-generation technology, it meets the millisecond-level real-time requirements of online monitoring of rotating machinery.

[0027] Other advantages, objectives and features of the present invention will become apparent in part from the following description, and in part from those skilled in the art through study and practice of the invention. Attached Figure Description

[0028] Figure 1 This is a flowchart of the present invention; Figure 2 This is a schematic diagram of an ideal representation of the signal to be analyzed in this invention; Figure 3 It is a time-domain plot drawn after adding Gaussian white noise to the original signal; Figure 4 The results are numerical simulations obtained without template comparison. Figure 5 The results are obtained from numerical simulations after template comparison. Detailed Implementation

[0029] The present invention will now be described in further detail so that those skilled in the art can implement it based on the description.

[0030] like Figure 1 As shown, this invention provides a time-frequency analysis method for frequency-modulated wavelet transform based on local entropy-selective scale extraction, comprising the following steps: Step 1) Obtain the sampling data of the target signal and generate a signal matrix according to the set sampling frequency and sampling time; Step 2) Generate a sliding Gaussian window function, and apply the sliding Gaussian window function to the signal matrix to generate a sliding window function matrix; Step 3) Discretize the frequency modulation angle range into multiple angle intervals, construct scaling basis functions for different angles based on frequency modulation wavelet transform, and match the scaling basis functions with the sliding window function matrix point by point to obtain the time-frequency sub-representation matrix for multiple angles; Step 4) Perform a Fourier transform on the time-frequency sub-representation matrix to generate a complex time-frequency representation matrix; Step 5) Calculate the partial derivatives of the time-frequency representation matrix and construct a redistribution operator based on the partial derivatives. Then, introduce the Dirichlet function to redefine the redistribution operator, re-express the partial derivatives, and constrain the phase derivatives. Then, extract time-frequency points with strong energy and stable phase changes from the original time-frequency matrix, and after dynamic adjustment, construct a time-frequency distribution matrix with good energy concentration characteristics. Step 6) Divide the time-frequency distribution matrix into several time-frequency blocks, calculate the Rényi entropy of each time-frequency block, and select the time-frequency block generated by the basis function angle with the smallest entropy value as the optimal time-frequency block; Step 7) Segment all the optimal time-frequency blocks to form a complete high-resolution time-frequency representation matrix.

[0031] This scheme addresses the issue of insufficient time-frequency representation accuracy for non-stationary signals by improving time-frequency resolution and energy concentration through multi-step processing. First, sampled data of the target signal is acquired, and a signal matrix is ​​generated according to a set sampling frequency and time. Then, a sliding Gaussian window function is constructed to process the signal matrix, forming a sliding window function matrix. Next, the frequency modulation angle range is discretized into multiple intervals, and scaling basis functions for different angles are constructed based on frequency modulation wavelet transform. These are then matched point-by-point with the sliding window function matrix to obtain a multi-angle time-frequency sub-representation matrix. A Fourier transform is performed on this matrix to generate a complex form time-frequency representation matrix. Then, partial derivatives of this matrix are calculated to construct a redistribution operator. A Dirichlet function is introduced to redefine the operator and re-express the partial derivatives, while constraining the phase derivative. Through this process, time-frequency points with strong energy and stable phase are extracted, and a time-frequency distribution matrix with high energy concentration is constructed after dynamic adjustment. The time-frequency distribution matrix is ​​divided into several time-frequency blocks, and the Rényi entropy of each block is calculated. The optimal time-frequency block corresponding to the minimum entropy value is selected. Finally, all optimal time-frequency blocks are concatenated to form a complete high-resolution time-frequency representation matrix.

[0032] Existing EMCT technology uses global Rényi entropy to optimize the basis function, which can improve the overall energy concentration, but ignores the differences in local time-frequency characteristics of the signal. For example, in mechanical shock signals, the frequency modulation characteristics of high-frequency transient components and low-frequency steady-state components are different, and global optimization cannot adaptively match local features, leading to residual energy diffusion. This invention effectively solves the local mismatch problem by dividing the time-frequency blocks and independently calculating the local entropy value, and selecting the optimal basis function angle for each time-frequency region.

[0033] In another approach, the time-frequency analysis method of frequency-modulated wavelet transform based on local entropy selection scale extraction, in step 1), generates a signal matrix according to the set sampling frequency and sampling time, and zero-padded both ends of the signal to generate an extended signal; In step 2), a sliding Gaussian window function is generated, and the extended signal is processed by the sliding Gaussian window function to generate a sliding window function matrix; The specific steps include: a. Based on the set sampling frequency f s and sampling time t tend Acquire target signal S ( t This generates the original time-domain signal matrix, which represents the sequence of sampling points; b. To avoid boundary effects, zero-padding is applied to both ends of the original time-domain signal matrix, with a padding length of [value missing]. N w / 2, generating extended signal S est ( t The expression is: S est ( t ) = [ s 1, s 2,…, s M ] in, M = f s ﹒t tend , where is the number of sampling points for the signal. N w The length of the Gaussian window function. s M For the original signal at the 1st M The value of each sampling point; c. Select a length of N w The Gaussian window function is used, and the overlap length of the Gaussian window function is set to... N ov Calculate the step size of the Gaussian window function. K 1= N w - N ov Next, calculate the number of windows that can be slidable. K = ( (N - K 1 ) / K 1), of which K Rounding down, the sliding Gaussian window function is expressed as: Where σ is the window function width factor; the larger the value, the wider the window and the smoother the window. N Original signal S ( t ) length; d. Using a sliding Gaussian window function, starting from the beginning of the extended signal, with a step size of... K 1. Slide the signal to cover the entire extended signal. Each slide extracts a local segment of the extended signal, multiplies it by the window function to obtain a weighted signal, and after traversal, generates a sliding window function matrix, expressed as: in, g ( t - t c ) as t c A sliding Gaussian window function centered at the center. t c Centered on time.

[0034] Vibration signal data (the target signal to be analyzed) is collected according to the set sampling frequency. An original vibration data matrix is ​​constructed within a specified sampling time, and a sliding Gaussian window function of a certain length is selected to form a sliding window matrix. Then, the sliding window is moved according to the set step size to gradually traverse the entire original vibration data matrix.

[0035] This scheme, after acquiring the target signal sampling data, uses a set sampling frequency... f s and sampling time t tend Generate the original time-domain signal matrix. To address time-frequency analysis distortion caused by signal boundary truncation, zero-padding is applied to both ends of the original signal matrix, with the padding length being the length of the Gaussian window function. N w Half of it, generating an extended signal S est ( t This extended signal includes the original signal point and zero-padding values ​​on both sides, ensuring that subsequent windowing processes cover the entire signal range. Select a length of... N w Gaussian window function, setting the window function overlap length N ov Calculate the window movement step size K 1= N w - N ov The number of sliding windows is calculated based on the total signal length and step size. K = ( (N - K1 ) / K 1) Round down. The Gaussian window function is used to round down from the starting point of the extended signal in steps of [number missing]. K 1. Sliding: Each time, a local segment of the extended signal is extracted and multiplied point-by-point with the window function to obtain a weighted signal. After traversing the entire signal, a sliding window function matrix is ​​formed. This matrix completely preserves the signal boundary information, providing a distortion-free basis for subsequent time-frequency analysis.

[0036] Existing technologies (such as classic STFT) directly window the raw signal, leading to energy leakage at the beginning and end of the signal due to incomplete coverage by the window function. For example, when analyzing a 0.1s mechanical impact signal, traditional methods exhibit frequency trajectory breaks and amplitude attenuation of up to 40% during the first and last 5ms. This implementation method achieves this through precise zero-fill length (… N w / 2) and window function parameter design, so that the boundary region obtains the same window function coverage as the signal center region. Experiments show that when processing bearing fault signals with 12.8kHz sampling and 1024-point window length, this method can still maintain the complete trajectory of characteristic frequency 125Hz within the first and last 10ms of the signal, and the amplitude error is controlled within 3%.

[0037] In another approach, the time-frequency analysis method based on frequency-modulated wavelet transform extracted by local entropy selection scale, in step 3), the construction of the scaling basis function discretizes the frequency modulation angle range [-π / 2, π / 2] into... N na Each angle interval is divided into several angle sequences, and each discrete angle is... η m The calculation formula is: The instantaneous frequency trajectories of a series of basis functions are calculated using the above formula, providing multi-angle rotation support for subsequent matching with the sliding window signal matrix.

[0038] Based on frequency-modulated wavelet transform, scaling basis functions at different angles are constructed. The expressions for the scaling basis functions are as follows: Among them, tan ( η m To adjust the amplitude of nonlinear frequency modulation, f For frequency, i The imaginary unit; By point-by-point matching of the scaling basis functions with the sliding window function matrix, a multi-angle time-frequency sub-representation matrix is ​​obtained. The expression for the time-frequency sub-representation matrix is ​​as follows: .

[0039] This scheme discretizes the frequency modulation angle range [-π / 2, π / 2] into... Nna An angle sequence is formed by several angle intervals, each discrete angle η m Through formula Calculated. Scaling basis functions at different angles are constructed based on frequency-modulated wavelet transform, and their expressions are as follows: , where tan ( η m This is used as the adjustment term for the nonlinear frequency modulation amplitude. The constructed scaling basis function and the sliding window function matrix are subjected to point-by-point matching operations to generate a multi-angle time-frequency sub-representation matrix, the specific expression of which is... This process achieves adaptive matching of nonlinear frequency modulation characteristics through angle discretization.

[0040] Existing technologies (such as fixed-angle SBCT) use a preset single frequency modulation angle basis function. When processing variable-speed bearing signals, a speed fluctuation exceeding 15% can cause instantaneous frequency trajectory breaks. For example, when analyzing a fault signal where the speed suddenly changes from 1800 rpm to 2200 rpm, traditional methods generate a frequency ambiguity band in the speed switching region, resulting in a decrease in the trajectory signal-to-noise ratio of approximately 12 dB. This implementation uses 32 discrete angles to cover the frequency modulation range of [-90°, 90°]. Experiments show that for wind turbine variable-speed signals with a fluctuation rate of up to 25%, a clear 125 Hz fault characteristic ridge can still be maintained in the speed abrupt change region, with a frequency tracking error of less than 0.5 Hz.

[0041] In another approach, the time-frequency analysis method based on local entropy-selective scale extraction using frequency-modulated wavelet transform, in step 4), the time-frequency representation matrix expression is: in, j The imaginary part of a complex number ， These are basis functions.

[0042] Time-frequency transformation of a signal is achieved by constructing scaling basis functions and performing multi-angle matching. First, the frequency modulation angle range is discretized into several parts, forming a set of basis function matrices, each basis function having a specific instantaneous frequency trajectory. Then, each basis function is matched one by one with a sliding window function matrix to capture the time-frequency characteristics of the signal at different angles. Through Fourier transform, the matching results are converted into the time-frequency domain, forming a three-dimensional sub-time-frequency representation containing different windows, different frequency points, and multiple frequency modulation angles.

[0043] In another approach, the time-frequency analysis method based on frequency-modulated wavelet transform extracted by local entropy selection scale, in step 5), calculates the partial derivative of the time-frequency representation matrix from step 4). The expression is: The redistribution operator is constructed based on partial derivatives, and its expression is: The redistribution operator is redefined using the Dirichlet function in SET, with the expression: Therefore, partial derivatives To further express: And it satisfies the following constraints: The effective time-frequency points with strong energy and stable phase changes are extracted from the original time-frequency representation matrix, expressed as follows: in, Delta For discrete frequency intervals, For a given time center t c and frequency center f c Below is the instantaneous frequency estimate obtained through the SET method; After dynamic adjustment, a time-frequency distribution matrix with energy concentration characteristics is constructed, expressed as: .

[0044] In the formula, This represents the optimized time-frequency distribution after synchronous extraction and transformation. It eliminates noise components outside the trajectory and significantly enhances the energy concentration on the instantaneous frequency trajectory.

[0045] By constructing a multi-angle basis function matching with the sliding window signal matrix, a matrix containing complex time-frequency representations is generated. Through filtering, noise components of non-instantaneous frequency trajectories are removed, and true energy points are retained, ultimately generating an optimized time-frequency distribution matrix.

[0046] This scheme calculates the partial derivatives of the complex time-frequency representation matrix in both time and frequency dimensions. Based on these partial derivatives, an initial redistribution operator is constructed to estimate the instantaneous frequency. The Dirichlet function is introduced to redefine and optimize this redistribution operator, constraining its output range. Simultaneously, the original partial derivative expression is transformed into a form associated with the optimized operator, and a strict phase derivative stationarity condition is imposed, limiting the phase change rate to no more than 0.1 radians per second. Valid time-frequency points are selected based on a dual criterion of signal energy intensity threshold and phase stationarity, with the energy threshold set at 5% of the maximum amplitude of the time-frequency matrix. Only time-frequency points that simultaneously satisfy the high energy and phase stationarity conditions are retained, and their instantaneous frequency estimates are calculated using a synchronous extraction transformation technique. Finally, a high-energy-density time-frequency distribution matrix is ​​dynamically constructed based on the selected high-quality time-frequency points.

[0047] Existing synchronous extraction and transformation techniques, when processing bearing outer ring fault signals under strong noise backgrounds, suffer from abnormal fluctuations in the trajectory at the characteristic frequency of 152Hz near the 0.15s point of impact due to the lack of a phase derivative constraint mechanism. The maximum instantaneous frequency deviation reaches 1.5Hz. This implementation, through strict phase change rate limitation and Dirichlet function optimization, analyzes the same 12.8kHz fault signal sampled for 0.5s. During the impact period, the 152Hz trajectory fluctuation amplitude is suppressed within ±0.15Hz, the artifact component caused by phase abrupt changes is reduced by approximately 90%, and the signal-to-noise ratio of the time-frequency plane characteristic frequency is improved by 18 dB.

[0048] In another approach, the time-frequency analysis method based on local entropy-selective scale extraction using frequency-modulated wavelet transform, in step 6), is based on the time-frequency distribution matrix... number of rows k The time-frequency distribution matrix is ​​divided into several time-frequency blocks, and the number of blocks in the vertical direction is determined. N nd Number of horizontal blocks N nc And calculate the number of rows in each time-frequency block. N h Number of columns N hh The calculation formula is: , If the number of rows in the time-frequency distribution matrix k If it is divisible by the number of blocks, use the result of the divisibility; otherwise, round down. from T e Extract the block matrix and evaluate its entropy value. T eTo obtain the three-dimensional matrix composed of the effective time-frequency point amplitudes under amplitude threshold and phase derivative constraints, the Rényi entropy formula is used: in, α Let be the order of Rényi entropy, with a value of 3. P i The energy distribution probability of the time-frequency module is calculated using the following formula: R e To further express: Next, each time-frequency block is traversed, the corresponding Rényi entropy value is calculated, and the time-frequency block generated by the basis function angle with the smallest entropy value is selected as the optimal time-frequency block.

[0049] Local entropy selection optimization aims to optimize the final time-frequency distribution by dividing the time-frequency distribution into blocks and evaluating entropy values ​​to select the sub-regions with the minimum entropy value. Entropy values ​​are used to quantify the concentration of information; the information corresponding to the minimum entropy value is more concentrated and can more accurately reflect the instantaneous frequency trajectory of the signal.

[0050] This scheme constructs a time-frequency distribution matrix based on its number of rows. k Divide the space into several time-frequency blocks. Determine the number of blocks in the vertical direction. N nd Number of horizontal blocks N nc Based on this, the number of rows in each time-frequency block is calculated. N h Number of columns N hh If the total number of rows in the matrix k Can be N nd If the value is divisible, use the integer value directly; otherwise, round down. Extract each submatrix from the matrix and use third-order Rényi entropy to evaluate the local energy distribution. When calculating the entropy, first solve for the sum of the squares of the energy values ​​of all points within the submatrix as the denominator, and use the square of the energy value of each point as the numerator to obtain the energy distribution probability of each point. P i Substitute the values ​​into the Rényi entropy formula to calculate the entropy value of each block. After traversing all time-frequency blocks to complete the entropy calculation, select the time-frequency block with the smallest Rényi entropy value as the optimal time-frequency block for the current local region. This optimal time-frequency block corresponds to the basis function angle that best matches the local signal characteristics.

[0051] The existing EMCT technology employs a global single Rényi entropy optimization strategy, which has significant limitations when analyzing complex gear fault signals containing transient impacts and steady-state vibrations. When a tooth breakage impact occurs at 0.3s, EMCT, due to its global optimization, cannot simultaneously adapt to the high-frequency transient characteristics of the impact segment and the low-frequency characteristics of the steady-state segment. This results in an energy diffusion rate as high as 42% for the 83Hz fault characteristic frequency trajectory during the 0.2s to 0.4s period, and the generation of a frequency ambiguity band at the impact point. This implementation divides the time-frequency plane of the 12.8kHz sampled signal for 1s into 20×20 blocks and independently calculates the local entropy value. During the 0.28s to 0.32s impact period, it automatically selects the basis function angle that matches the transient characteristics, and adapts to the steady-state characteristics in other periods. The processing results show that the 83Hz characteristic trajectory remains continuous throughout the entire time period, the energy concentration at the impact point reaches 95%, which is 53 percentage points higher than EMCT, and the background noise in the time-frequency plane is reduced by 22 dB.

[0052] In another approach, the time-frequency analysis method based on local entropy-selective scale extraction using frequency-modulated wavelet transform, in step 7), concatenates all optimal time-frequency blocks to form a complete high-resolution time-frequency representation matrix, expressed as: .

[0053] In another approach, the time-frequency analysis method based on frequency-modulated wavelet transform extracted by local entropy selection of scale, after step 7), also includes: Step 8) Reassignment operator based on Dirichlet function The complete high-resolution time-frequency representation matrix TFPR opt ( f c , t c The data is redistributed into a new time-frequency representation matrix, expressed as: .

[0054] Through the above steps, the time-frequency distribution can be finely optimized, significantly improving the clarity and readability of the time-frequency representation, thereby accurately locating the characteristic positions of the impact components.

[0055] Numerical simulation: In numerical simulation, the target signal The structure consists of multiple frequency modulations FM The superposition of components, each with a complex instantaneous frequency trajectory, is used to simulate the nonlinear and multi-component characteristics that may occur in real signals. The specific construction formula is as follows: The instantaneous frequency trajectories of each component are as follows: Its ideal time-frequency characterization is as follows Figure 2 As shown, the target signal is sampled at a frequency... f s = 240Hz, sampling time t tend = 4s, construct the signal components in the above formula and superimpose them to form the target signal. S ( t To simulate a real-world noise environment, Gaussian white noise was added to the signal, with the signal-to-noise ratio (SNR) set to 10 dB. Figure 3 This is a time-domain plot of the original signal after adding Gaussian white noise. The plot shows the target signal with added noise. S ( t The time-domain waveform of the waveform is obtained. Plot all instantaneous frequency components. f 1 ( t ), f 2 ( t ), ..., f 6 ( t This is used to verify the frequency complexity of the target signal. (Sliding Gaussian window length) N w = 200, sliding step size N ov = 199, Discrete number of basis function rotation angles N na = 70, the entropy selection block parameter is set to: number of vertical blocks N nd = 20, number of horizontal blocks N nc = 10, frequency smoothness conditional threshold b = 20 Amplitude filtering threshold a = 0.02, then the optimized time-frequency representation can be generated. .

[0056] The numerical simulation results obtained using the technical solution of this invention are as follows: Figure 5 As shown, the energy points on the instantaneous frequency trajectory of the signal are clearly concentrated, and the blurred areas are significantly reduced. Noise components are effectively suppressed, and false time-frequency points are almost completely eliminated. Compared with the time-frequency analysis results without template matrix comparison similarity analysis, as shown... Figure 4 As shown, it is evident that spurious components are generated between components, which significantly causes severe interference and affects the readability of the time-frequency representation. Simulations demonstrate that the time-frequency analysis method based on local entropy-selective scale extraction of the present invention has significant advantages in the time-frequency representation of multi-component nonlinear signals. It not only preserves the true frequency components but also effectively removes noise and spurious components, significantly improving the readability and resolution of the time-frequency distribution.

[0057] In another approach, the time-frequency analysis method based on local entropy-selective scale extraction using frequency-modulated wavelet transform, step 7) includes the following steps in the splicing process: A. Boundary parameter setting: Define the boundary time points of adjacent time-frequency blocks t b = m × N h × △ t ; in, m The time-frequency block number, 1 ≤ m≤N nd - 1, △ t For time resolution, △ t = 1 / f s ; Calculate the number of overlapping points ; Calculate the overlap time half-width ; B. Window function construction: In the global time coordinate t interval [ t b - Δ T , t b + Δ T Within, according to time resolution △ t Values ​​are taken at equal intervals; Construct the left window function: Construct the right window function: C. Fusion computing: For global time coordinates t and frequency index p Perform fusion computing: Where, 1 ≤ p ≤ N f , , B m ( t , p ) is the first m The optimal time-frequency block is at point ( t , p The value at ) B m+1 ( t, p ) is the first m+ One optimal time-frequency block at point ( t , p The value at () D. Matrix construction: Generate the final time-frequency matrix: .

[0058] This scheme first acquires the target signal to generate a signal matrix and performs zero-filling expansion. A sliding Gaussian window function is constructed to process the expanded signal and form a window function matrix. The frequency modulation angle is discretized to construct a scaling basis function, which is matched with the window function matrix to generate a multi-angle time-frequency sub-representation matrix. After obtaining the complex time-frequency matrix through Fourier transform, partial derivatives are calculated to construct a redistribution operator. A Dirichlet function optimization operator is introduced, and the phase derivative is constrained. High-energy and phase-stable time-frequency points are extracted to construct a time-frequency distribution matrix. The matrix is ​​divided into several time-frequency blocks, and the optimal block with the smallest entropy value is selected after calculating the third-order Rényi entropy of each block. Boundary smoothing is performed when assembling the optimal blocks: the boundary time points of adjacent blocks are defined. t b = m × N h × △ t , where △ t = 1 / f s .

[0059] Calculate the number of overlapping points Thus, the overlap time half-width is obtained. In the interval [ t b -Δ T , t b + Δ T Construct left and right window functions within the global time coordinates. t and frequency index p A fusion calculation is performed to ultimately construct a seamless time-frequency matrix.

[0060] Existing technologies employ a rigid splicing strategy. When processing wind turbine bearing speed change signals (speed increasing from 1500 rpm to 2000 rpm within 0.5 seconds), a step abrupt change in amplitude of 0.3 Hz occurs in the characteristic frequency 125 Hz trajectory at the block boundary, resulting in an energy distortion rate of 18% in the boundary region. This implementation effectively solves this problem through a gradual window function fusion mechanism.

[0061] Example 1 The fault signal (characteristic frequency 112Hz) of the inner ring of a variable speed fan bearing was analyzed. The sampling frequency was 12.8kHz, the duration was 1s, and the signal was divided into 20 time-frequency blocks. After boundary smoothing, the continuity of the 112Hz trajectory throughout the entire time period reached 99.8%, the fluctuation amplitude at the boundary of the speed switching point at 0.35s was only ±0.02Hz, and the energy transition smoothness between adjacent blocks exceeded 97%.

[0062] Comparative Example 1 The same signals were directly spliced ​​together. Obvious trajectory jumps appeared at the block boundary points such as 0.32s and 0.37s, with a maximum jump amplitude of 0.5Hz. The energy distortion rate in the boundary region was 22.7%. The trajectory signal-to-noise ratio was reduced by 18dB compared to Example 1 during the 0.3s to 0.4s period.

[0063] The comparison shows that the present invention achieves smooth energy transition by constructing a time-domain gradient window, reducing the frequency trajectory discontinuity at the block boundary by more than 85%, and significantly improving the reliability of time-frequency representation of variable operating condition signals.

[0064] In another approach, the time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform further includes a phase continuity calibration step after step 6) and before step 7), which includes: S1. Determine the boundary between adjacent blocks: For two adjacent optimal time-frequency blocks generated in step 6), identify their natural intersection time point as the boundary time point. t b ; S2. Calculate the boundary frequency derivative: Calculate the previous optimal time-frequency block TFPR prev ( f , t At the boundary time point t b Right derivative of instantaneous frequency at point: ωL Calculate the next optimal time-frequency block TFPR next ( f , t At the boundary time point t b Left derivative ωR of instantaneous frequency at point: S3. Dynamic phase compensation: If the absolute value of the derivative difference |ωL-ωR| > preset tolerance threshold ε Then, a complex domain phase rotation operation is performed on the next optimal time-frequency block to generate a compensated time-frequency block. TFPR comp Its expression is: in, t For time variables, f For frequency variables, l It is the imaginary unit and satisfies l 2 = -1; S4. Replace the compensation block: Replace the compensated time-frequency block TFPR comp ( t , f This is used as the new next optimal time-frequency block.

[0065] The core process of this scheme first acquires the target signal sampling data and generates a signal matrix according to the set sampling frequency and sampling time. Next, a sliding Gaussian window function is constructed to process the signal matrix and form a window function matrix. Then, the frequency modulation angle range is discretized, and corresponding scaling basis functions are constructed and matched with the window function matrix to generate a multi-angle time-frequency sub-representation matrix. A Fourier transform is performed on this matrix to obtain a complex time-frequency representation matrix. Then, the partial derivatives of this matrix are solved to construct a redistribution operator. A Dirichlet function optimization operator is introduced, and the rate of change of the phase derivative is constrained. Based on this, high-energy and phase-stable time-frequency points are extracted to construct a time-frequency distribution matrix. This matrix is ​​divided into several time-frequency blocks, and the Rényi entropy value of each block is calculated independently. The optimal time-frequency block with the smallest entropy value is selected. Before splicing all the optimal time-frequency blocks to form a complete time-frequency matrix, a phase continuity calibration step is performed: the natural intersection time point of adjacent optimal time-frequency blocks is determined as the boundary time point. The right derivative of the instantaneous frequency of the previous block at the boundary and the left derivative of the instantaneous frequency of the next block at the boundary are calculated. If the absolute value of the difference between the two exceeds a preset tolerance threshold, a complex domain phase rotation operation is performed on the latter block to generate a compensation block, which replaces the original latter time-frequency block. Finally, all optimal blocks are spliced ​​together to form the final high-resolution time-frequency representation matrix.

[0066] Existing technologies directly splice time-frequency blocks without considering the consistency of instantaneous frequency derivatives at the boundaries of adjacent blocks. For example, when analyzing composite fault signals during gearbox acceleration, phase jumps easily occur in the characteristic frequency trajectories between adjacent blocks at points of sudden speed change, resulting in a 0.5Hz deviation in the instantaneous frequency estimate at the boundary, which seriously affects the continuous identification of fault features.

[0067] Example 2 Analysis of composite fault signals during the acceleration process of a gearbox reveals the characteristic frequency of inner race damage. f 1 = 85Hz, characteristic frequency of outer ring damage f 2 = 45Hz, sampling frequency 12.8kHz, signal duration 3s. Measured before calibration at the boundary of the 1.5s speed change abrupt change point. f The right derivative is +3.2 Hz / s. fThe left derivative is +1.8 Hz / s. After phase rotation compensation, the boundary region... f The maximum deviation of trajectory 1 is 0.15Hz. f The maximum deviation of the trajectory is 0.08 Hz, and the smoothness of energy transition between adjacent blocks reaches 98.7%.

[0068] Comparative Example 2 Phase calibration is omitted for identical signals. At the 1.5s boundary point, f Trajectory 1 shows a 0.48Hz step break. f Trajectory 2 shows a 0.33Hz offset. The average distortion rate of the instantaneous frequency estimate in the boundary region reaches 18%, resulting in aliasing of fault characteristics and making it impossible to effectively separate the inner and outer damage components.

[0069] The technical solution of the present invention is as follows: Step (1): Construct the sliding window function matrix of the signal. Using the acquired target signal data, generate a signal matrix based on the sampling frequency and sampling time. Zero-padding is applied to both ends of the signal data to avoid missing boundary data. Then, a sliding Gaussian window function is used to slide across the signal matrix with a certain step size, generating a sliding window function matrix covering the entire signal, which serves as the input for subsequent time-frequency analysis.

[0070] Step (2): Generate scaling basis functions and perform multi-angle matching. Discretize the frequency modulation angle into several parts (e.g., (Intervals) are used to construct basis functions at different angles. These basis functions are then used to perform point-by-point matching on the sliding window function matrix to obtain the time-frequency sub-representation matrix for each angle. The matching results are then transformed into the time-frequency domain using Fourier transform, forming a complex time-frequency representation matrix, which provides the foundation for subsequent energy extraction.

[0071] Step (3): Based on the complex time-frequency representation matrix obtained in step (2), effective time-frequency points are selected using the amplitude threshold and phase stability conditions. The energy distribution of the time-frequency points is concentrated in the region consistent with the instantaneous frequency trajectory, and ambiguity components are eliminated, thereby enhancing the clustering of the time-frequency representation.

[0072] Step (4): Divide the enhanced time-frequency matrix obtained in step (3) into several small blocks, calculate the Rényi entropy for each block, and select the basis function angle with the smallest entropy value as the optimal matching angle. For each small block, extract the optimized data corresponding to that angle to ensure that the energy of that block is most concentrated in the time-frequency representation.

[0073] Step (5): Based on the multiple time-frequency blocks obtained in step (4), all time-frequency blocks that have undergone entropy optimization are spliced ​​together to form a complete high-resolution time-frequency representation matrix. This matrix can accurately describe the instantaneous frequency trajectory of the signal, while significantly improving the readability and accuracy of the signal's time-frequency representation.

[0074] This method combines the flexibility of SBCT, the energy optimization capability of EMCT, and the post-processing accuracy of SET. The time-frequency analysis method based on local entropy-selective scale extraction using frequency-modulated wavelet transform further improves the resolution and accuracy of time-frequency representation by jointly adjusting the basis function design, entropy optimization criteria, and post-processing strategies, providing a novel approach for the accurate characterization of multi-component non-stationary signals.

[0075] This method is applicable to the analysis of multi-component non-stationary signals, including but not limited to mechanical vibration signals, radar echo signals, or biomedical signals.

[0076] Although the embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments. They can be applied to various fields suitable for the present invention. For those skilled in the art, other modifications can be easily made. Therefore, without departing from the general concept defined by the claims and their equivalents, the present invention is not limited to the specific details.

Claims

1. A time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform, characterized in that, Includes the following steps: Step 1) Obtain the sampling data of the target signal and generate a signal matrix according to the set sampling frequency and sampling time; Step 2) Generate a sliding Gaussian window function, and apply the sliding Gaussian window function to the signal matrix to generate a sliding window function matrix; Step 3) Discretize the frequency modulation angle range into multiple angle intervals, construct scaling basis functions for different angles based on frequency modulation wavelet transform, and match the scaling basis functions with the sliding window function matrix point by point to obtain the time-frequency sub-representation matrix of multiple angles; Step 4) Perform a Fourier transform on the time-frequency sub-representation matrix to generate a complex time-frequency representation matrix; Step 5) Calculate the partial derivatives of the time-frequency representation matrix and construct a redistribution operator based on the partial derivatives. Then, introduce the Dirichlet function to redefine the redistribution operator, re-express the partial derivatives, and constrain the phase derivatives. Then, extract time-frequency points with strong energy and stable phase changes from the original time-frequency matrix, and after dynamic adjustment, construct a time-frequency distribution matrix with good energy concentration characteristics. Step 6) Divide the time-frequency distribution matrix into several time-frequency blocks, calculate the Rényi entropy of each time-frequency block, and select the time-frequency block generated by the basis function angle with the smallest entropy value as the optimal time-frequency block; Step 7) Segment all the optimal time-frequency blocks to form a complete high-resolution time-frequency representation matrix.

2. The time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform as described in claim 1, characterized in that, In step 1), a signal matrix is ​​generated according to the set sampling frequency and sampling time, and zero-padding is performed at both ends of the signal to generate an extended signal; In step 2), a sliding Gaussian window function is generated, and the extended signal is processed by the sliding Gaussian window function to generate a sliding window function matrix; The specific steps include: a. Based on the set sampling frequency f s and sampling time t tend Acquire target signal S ( t ), generating the original time-domain signal matrix; b. Zero-padded both ends of the original time-domain signal matrix, with a padding length of... N w / 2, generating extended signal S est ( t The expression is: S est ( t ) = [ s 1, s 2,…, s M ] in, M = f s ﹒t tend , where is the number of sampling points for the signal. N w The length of the Gaussian window function. s M For the original signal at the 1st M The value of each sampling point; c. Select a length of N w The Gaussian window function is used, and the overlap length of the Gaussian window function is set to... N ov Calculate the step size of the Gaussian window function. K 1 = N w - N ov Next, calculate the number of windows that can be slidable. K = ( (N - K 1 ) / K 1), of which K Rounding down, the sliding Gaussian window function is expressed as: Where σ is the window function width factor, N Original signal S ( t ) length; d. Using a sliding Gaussian window function, starting from the beginning of the extended signal, with a step size of... K 1. Slide the signal to cover the entire extended signal. Each slide extracts a local segment of the extended signal, multiplies it by the window function to obtain a weighted signal, and after traversal, generates a sliding window function matrix, expressed as: in, g ( t - t c ) as t c A sliding Gaussian window function centered at the center. t c Centered on time.

3. The time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform as described in claim 2, characterized in that, In step 3), the frequency modulation angle range [-π / 2, π / 2] is discretized into... N na Each angle interval is divided into several angle sequences, and each discrete angle is... η m The calculation formula is: Based on frequency-modulated wavelet transform, scaling basis functions at different angles are constructed. The expressions for the scaling basis functions are as follows: Among them, tan ( η m To adjust the amplitude of nonlinear frequency modulation, f For frequency, i The imaginary unit; By point-by-point matching of the scaling basis functions with the sliding window function matrix, a multi-angle time-frequency sub-representation matrix is ​​obtained. The expression for the time-frequency sub-representation matrix is ​​as follows: 。 4. The time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform as described in claim 3, characterized in that, In step 4), the time-frequency representation matrix expression is: in, j The imaginary part of a complex number ， These are basis functions.

5. The time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform as described in claim 4, characterized in that, In step 5), the partial derivatives of the time-frequency representation matrix from step 4) are calculated. The expression is: The redistribution operator is constructed based on partial derivatives, and its expression is: The redistribution operator is redefined using the Dirichlet function in SET, with the expression: Therefore, partial derivatives To further express: And it satisfies the following constraints: The effective time-frequency points with strong energy and stable phase changes are extracted from the original time-frequency representation matrix, expressed as follows: in, Delta For discrete frequency intervals, For a given time center t c and frequency center f c Below is the instantaneous frequency estimate obtained through the SET method; After dynamic adjustment, a time-frequency distribution matrix with energy concentration characteristics is constructed, expressed as: 。 6. The time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform as described in claim 5, characterized in that, In step 6), based on the time-frequency distribution matrix... number of rows k The time-frequency distribution matrix is ​​divided into several time-frequency blocks, and the number of blocks in the vertical direction is determined. N nd Number of horizontal blocks N nc And calculate the number of rows in each time-frequency block. N h Number of columns N hh The calculation formula is: , If the number of rows in the time-frequency distribution matrix k If it is divisible by the number of blocks, use the result of the divisibility; otherwise, round down. from T e Extract the block matrix and evaluate its entropy value. T e To obtain the three-dimensional matrix composed of the effective time-frequency point amplitudes under amplitude threshold and phase derivative constraints, the Rényi entropy formula is used: in, α Let be the order of Rényi entropy, with a value of 3. P i The energy distribution probability of the time-frequency module is calculated using the following formula: R e To further express: Next, each time-frequency block is traversed, the corresponding Rényi entropy value is calculated, and the time-frequency block generated by the basis function angle with the smallest entropy value is selected as the optimal time-frequency block.

7. The time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform as described in claim 6, characterized in that, In step 7), all optimal time-frequency blocks are concatenated to form a complete high-resolution time-frequency representation matrix, expressed as: .

8. The time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform as described in claim 7, characterized in that, After step 7), it also includes: Step 8) Reassignment operator based on Dirichlet function The complete high-resolution time-frequency representation matrix TFPR opt ( f c , t c The data is redistributed into a new time-frequency representation matrix, expressed as: 。 9. The time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform as described in claim 6, characterized in that, Step 7) includes the following steps in the splicing process: A. Boundary parameter setting: Define the boundary time points of adjacent time-frequency blocks t b = m × N h × △ t ; in, m The time-frequency block number, 1 ≤ m ≤ N nd - 1, △ t For time resolution, △ t = 1 / f s ; Calculate the number of overlapping points ; Calculate the overlap time half-width ; B. Window function construction: In the global time coordinate t interval [ t b -Δ T , t b +Δ T Within, according to time resolution △ t Values ​​are taken at equal intervals; Construct the left window function: Construct the right window function: C. Fusion computing: For global time coordinates t and frequency index p Perform fusion computing: Where, 1 ≤ p ≤ N f , , B m ( t , p ) is the first m The optimal time-frequency block is at point ( t , p The value at ) B m+1 ( t , p ) is the first m+ One optimal time-frequency block at point ( t , p The value at () D. Matrix construction: Generate the final time-frequency matrix: 。 10. The time-frequency analysis method based on local entropy-selective scale extraction of frequency-modulated wavelet transform as described in claim 1, characterized in that, After step 6) and before step 7), a phase continuity calibration step is also included, which includes: S1. Determine the boundary between adjacent blocks: For two adjacent optimal time-frequency blocks generated in step 6), identify their natural intersection time point as the boundary time point. t b ; S2. Calculate the boundary frequency derivative: Calculate the previous optimal time-frequency block TFPR prev ( f , t At the boundary time point t b Right derivative of instantaneous frequency at point: ωL Calculate the next optimal time-frequency block TFPR next ( f , t At the boundary time point t b Left derivative ωR of the instantaneous frequency at point: S3. Dynamic phase compensation: If the absolute value of the derivative difference |ωL-ωR| > preset tolerance threshold ε Then, a complex domain phase rotation operation is performed on the next optimal time-frequency block to generate a compensated time-frequency block. TFPR comp Its expression is: in, t For time variables, f For frequency variables, l It is the imaginary unit and satisfies l 2 = -1; S4. Replace the compensation block: Replace the compensated time-frequency block TFPR comp ( t , f This is used as the new next optimal time-frequency block.

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