Instantaneous frequency identification method based on local maximum synchronous extrusion generalized s transform algorithm

Abstract

The application provides a local maximum synchronous extrusion generalized S transform algorithm-based instantaneous frequency identification method, which is composed of a generalized S transform, a local maximum synchronous extrusion operator and a local modulus maximum value ridge extraction algorithm. Specifically, the method estimates the instantaneous frequency through time-frequency coefficients of the generalized S transform, rearranges the time-frequency coefficients through the local maximum synchronous extrusion operator, and finally extracts the instantaneous frequency curve through the local modulus maximum value method. The method can effectively identify the instantaneous frequency of a time-varying structure.

Description

Technical Field

[0001] This embodiment relates to the field of structural health monitoring and safety assessment technology, and in particular to an instantaneous frequency identification method based on the local maximum synchronous compression generalized S-transform algorithm. Background Technology

[0002] During service, civil engineering structures are frequently subjected to external forces of varying frequencies and amplitudes, such as those from the natural environment and working loads, resulting in vibrations and deformations. Therefore, accurate monitoring and analysis of the structural health status are essential to ensure its safety and reliability. Instantaneous frequency, as a crucial modal parameter reflecting the structural state, is of great significance for structural safety assessment, damage detection, and health monitoring. However, due to interference from external factors and the inherent nonlinear and time-varying characteristics of the structure, the structural response is generally a non-stationary signal. This makes the accurate identification of the structure's instantaneous frequency a pressing issue that needs to be addressed.

[0003] Time-frequency analysis methods can simultaneously obtain the changes of signals in both the time and frequency domains, and can also more comprehensively describe the short-term bursts and long-term stable characteristics of signals, making them highly suitable for the analysis of non-stationary signals. The generalized S-transform, as a time-frequency analysis method, not only inherits the linear characteristics of the short-time Fourier transform and eliminates the interference term problem of the Wigner-Ville distribution, but also possesses multi-scale analysis capabilities similar to continuous wavelet transforms. Therefore, it is often used in the identification of instantaneous frequencies in time-varying structures. However, the generalized S-transform has low time-frequency resolution and cannot accurately extract instantaneous frequency curves. Therefore, how to improve the identification accuracy of the generalized S-transform still requires further research. Summary of the Invention

[0004] To address the shortcomings and deficiencies of existing technologies, this invention provides a local maximum synchronous squeeze generalized S-transform algorithm. This algorithm improves the time-frequency resolution and recognition accuracy of existing generalized S-transform algorithms by estimating the instantaneous frequency of the time-frequency coefficients of the generalized S-transform and combining it with the local maximum synchronous squeeze operator.

[0005] This method consists of three parts: a generalized S-transform, a local maximum synchronizing squeeze operator, and a local modulus maxima ridge extraction algorithm. Specifically, the instantaneous frequency is first estimated using the time-frequency coefficients of the generalized S-transform; then, the time-frequency coefficients are rearranged using the local maximum synchronizing squeeze operator; and finally, the instantaneous frequency curve is extracted using the local modulus maxima method. This method can effectively identify the instantaneous frequency of time-varying structures.

[0006] The following technical solution is specifically adopted in this embodiment:

[0007] A method for instantaneous frequency identification based on the local maximum synchronous squeeze generalized S-transform algorithm is characterized by: estimating the instantaneous frequency using the time-frequency coefficients of the generalized S-transform, rearranging the time-frequency coefficients using the local maximum synchronous squeeze operator, and finally extracting the instantaneous frequency curve using the local modulus maxima method.

[0008] Furthermore, the specific steps include:

[0009] Step S1: Perform a generalized S-transform on the signal to obtain the time-frequency diagram;

[0010] Step S2: Estimate the instantaneous frequency based on the time-frequency coefficients of the generalized S-transform;

[0011] Step S3: Search for the maximum value of the time-frequency coefficient modulus by setting a sliding window and construct the frequency band;

[0012] Step S4: Rearrange the time-frequency coefficients within the frequency band to the instantaneous frequency position corresponding to the maximum value of the time-frequency coefficient modulus;

[0013] Step S5: Use the local modulus maxima method to extract the instantaneous frequency curve.

[0014] Furthermore, step S1 specifically includes:

[0015] The generalized S-transform of the signal is expressed as follows:

[0016]

[0017] In the formula, S(τ,f) are the time-frequency coefficients of the generalized S-transform, τ is the parameter for adjusting the position of the Gaussian window on the time axis, f is the frequency, t is the time, and e is the frequency. -i2πft This is the phase correction factor, where i is the imaginary unit;

[0018] h(τ-t,f) is a Gaussian window function, and its expression is:

[0019]

[0020] In the formula, m and n are parameters for adjusting the duration and decay rate of the Gaussian window;

[0021] Accordingly, the frequency domain expression of S(τ,f) is:

[0022]

[0023] In the formula, X(f+α) is the Fourier transform of the signal x(t), and its expression is:

[0024]

[0025] Furthermore, step S2 specifically includes:

[0026] Since any signal can be approximated by adding simple harmonic waves, consider the harmonic signal x(t) = A(t)cos(2πf0t) and perform its Fourier transform:

[0027]

[0028] In the formula, δ{·} is the Kronecker function;

[0029] Therefore, S(τ,f) can be rewritten as:

[0030]

[0031] Taking the partial derivative of S(τ,f) with respect to τ, we get:

[0032]

[0033] At this moment, the instantaneous frequency ω(τ,f) of x(t) is expressed as:

[0034]

[0035] Furthermore, step S3 specifically includes:

[0036] Set a local sliding window with a width of a in the frequency direction, and then search for the maximum value of the S(τ,f) modulus in the local sliding window at each time. Retain the time-frequency coefficients in the window to construct the frequency band and set the coefficient values ​​outside the frequency band to zero, as shown in Equation (9).

[0037]

[0038] Then, according to equation (10), retain the instantaneous frequency corresponding to β, and set the rest of the positions to zero;

[0039]

[0040] Further, the process of step S4 is as shown in equation (11), in which the retained time-frequency coefficients are squeezed and rearranged in the frequency direction to obtain a clear time-frequency diagram of the local maximum synchronic squeezed generalized S-transform.

[0041]

[0042] Furthermore, the implementation process of step S5 is as follows:

[0043] First, search for the modulus maxima of T(τ,γ) within the frequency range at each time step, and record the location. The expression for this is:

[0044] P(τ,γ)=max{|T(τ,γ)|} (12)

[0045] Then, the obtained positions are converted into corresponding instantaneous frequencies according to the following formula and connected together in time series to obtain the instantaneous frequency curve;

[0046]

[0047] In the formula, q is the maximum value of the frequency axis of the time-frequency diagram, and r is the dimension of the coefficient matrix of the local maximum synchronous squeeze generalized S-transform.

[0048] Compared to existing technologies, this invention and its preferred embodiment improve the time-frequency resolution and recognition accuracy of existing generalized S-transform algorithms by estimating the instantaneous frequency of the time-frequency coefficients of the generalized S-transform and combining it with a local maximum synchronization squeeze operator. It can effectively identify the instantaneous frequency of time-varying structures. Attached Figure Description

[0049] The following detailed description of this embodiment, in conjunction with the accompanying drawings and specific implementation details, is as follows:

[0050] Figure 1 This is a flowchart of a method according to an embodiment of the present invention.

[0051] Figure 2 This is a diagram of the experimental apparatus in an embodiment of the present invention.

[0052] Figure 3 This is a diagram of the measured acceleration response signal in an embodiment of the present invention.

[0053] Figure 4 This is a diagram of the first-order component signal in an embodiment of the present invention.

[0054] Figure 5 This is the time-frequency diagram of the generalized S-transform in an embodiment of the present invention.

[0055] Figure 6 This is the time-frequency diagram of the generalized S-transform of the local maximum synchronous compression in this embodiment of the invention.

[0056] Figure 7 This is a diagram showing the instantaneous frequency identification result of the acceleration response signal in an embodiment of the present invention. Detailed Implementation

[0057] To make the features and advantages of this patent more apparent and understandable, specific embodiments are provided below, along with accompanying drawings, for detailed explanation:

[0058] like Figure 1 As shown, the specific steps of the method provided in this embodiment of the invention are as follows:

[0059] Step 1) The generalized S-transform can effectively reflect the short-term burst conditions and long-term stability characteristics of a signal. Therefore, the generalized S-transform is first performed on the signal, and its expression is:

[0060]

[0061] In the formula, S(τ,f) are the time-frequency coefficients of the generalized S-transform, τ is the parameter for adjusting the position of the Gaussian window on the time axis, f is the frequency, t is the time, and e is the frequency. -i2πft is the phase correction factor, and i is the imaginary unit.

[0062] h(τ-t,f) is a Gaussian window function, and its expression is:

[0063]

[0064] In the formula, m and n are parameters for adjusting the duration and decay rate of the Gaussian window.

[0065] Accordingly, the frequency domain expression of S(τ,f) is:

[0066]

[0067] In the formula, X(f+α) is the Fourier transform of the signal x(t), and its expression is:

[0068]

[0069] Step 2) Estimate the instantaneous frequency based on the time-frequency coefficient.

[0070] Since any signal can be approximated by adding simple harmonic waves, for ease of calculation, consider the harmonic signal x(t) = A(t)cos(2πf0t), and perform a Fourier transform on it to obtain...

[0071]

[0072] In the formula, δ{·} is the Kronecker function.

[0073] Therefore, S(τ,f) can be rewritten as

[0074]

[0075] Taking the partial derivative of S(τ,f) with respect to τ, we can obtain...

[0076]

[0077] At this point, the instantaneous frequency ω(τ,f) of x(t) can be expressed as:

[0078]

[0079] Step 3) Set the local sliding window to search for the maximum value of the modulus of the frequency coefficients and construct the frequency band.

[0080] Set a local sliding window with a width of a in the frequency direction, then search for the maximum value of the S(τ,f) modulus in the local sliding window at each time step, and finally retain the time-frequency coefficients in the window to construct the frequency band and set the coefficient values ​​outside the frequency band to zero. Its expression is shown in Equation (9).

[0081]

[0082] Then, retaining the instantaneous frequency corresponding to β and setting the rest to zero, the equation can be expressed as follows:

[0083]

[0084] Step 4) Rearrange the time-frequency coefficients within the frequency band to the instantaneous frequency position corresponding to the maximum value of the time-frequency coefficient modulus.

[0085] As shown in Equation (11), by squeezing and rearranging the retained time-frequency coefficients in the frequency direction, a clear time-frequency diagram of the local maximum synchronicity squeezed generalized S-transform can be obtained.

[0086]

[0087] Step 5) Extract the instantaneous frequency curve using the local modulus maxima method.

[0088] To further extract accurate dynamic characteristics of the signal from the time-frequency plot and reduce noise interference, the local maxima method of time-frequency coefficient modulus is used to extract the instantaneous frequency curve based on the identification results of the local maximum synchronous squeeze transformation. The implementation steps of this method are as follows:

[0089] First, search for the modulus maxima of T(τ,γ) within the frequency range at each time step and record the location. The expression for this is:

[0090] P(τ,γ)=max{|T(τ,γ)|} (12)

[0091] Then, the obtained positions are converted into corresponding instantaneous frequencies according to the following formula and connected together according to the time sequence to obtain the instantaneous frequency curve.

[0092]

[0093] In the formula, q is the maximum value of the frequency axis of the time-frequency diagram, and r is the dimension of the coefficient matrix of the local maximum synchronous squeeze generalized S-transform.

[0094] like Figures 2-6 As shown, the following is a specific embodiment example to further illustrate the solution.

[0095] Step 1) Select a uniform cross-section aluminum alloy cantilever beam with a length of 500mm, a cross-sectional size of 40mm × 15mm, and a weight of 0.81kg. Weld an aluminum alloy plate of size [size missing] to the fixed end and fix the component to the reaction frame with bolts. Place a 1kg mass block at the cantilever end and place a magnet above it. Figure 2 As shown.

[0096] Step 2) Based on the "freezing method", the fundamental frequencies of the cantilever beam with and without mass blocks were measured to be 21.24 Hz and 47.06 Hz, respectively.

[0097] Step 3) Apply excitation to the end of the cantilever beam using a Donghua LC-02 hammer with a steel hammerhead. At 2 seconds, use a magnet to lift the mass block pre-placed at the cantilever end, changing the structure's mass and thus altering the cantilever beam's natural frequency. Finally, strike the cantilever beam again at 2.3 seconds to prevent the response signal from decaying too quickly. Set the sampling frequency to 2kHz, and then use a Donghua DH118E accelerometer and a Donghua DH5299N dynamic data acquisition instrument to collect acceleration data at the mid-span of the cantilever beam throughout the entire time history. Figure 3 As shown.

[0098] Step 4) Decompose the response signal using analytical mode decomposition to obtain the first-order component signal, such as... Figure 4 As shown.

[0099] Step 5) Perform a generalized S-transform on the first-order component acceleration response signal to obtain the time-frequency diagram as shown below. Figure 5 As shown.

[0100] Step 6) Rearrange the time-frequency coefficients of the generalized S-transform using the local maximum synchronization squeeze operator, such as... Figure 6 As shown.

[0101] Step 7) Extract the instantaneous frequency curve using the local modulus maxima method, such as... Figure 7 As shown.

[0102] This patent is not limited to the above-described preferred embodiments. Anyone can derive other forms of instantaneous frequency identification methods based on the local maximum synchronous compression generalized S-transform algorithm under the guidance of this patent. All equivalent changes and modifications made in accordance with the scope of this patent application should be covered by this patent.

Claims

1. A method for instantaneous frequency identification based on local maximum synchronous extrusion generalized S-transform algorithm, characterized in that: The instantaneous frequency is estimated by time-frequency coefficients of generalized S transform, the time-frequency coefficients are rearranged by local maximum synchronous squeezing operator, and finally the instantaneous frequency curve is extracted by local modulus maximum method; Specifically comprising the following steps: Step S1: performing generalized S transform on the signal to obtain a time-frequency graph; Step S2: estimating the instantaneous frequency according to the time-frequency coefficients of the generalized S transform; Step S3: searching the modulus maximum value of the time-frequency coefficients by setting a sliding window and constructing a frequency band; Step S4: rearranging the time-frequency coefficients in the frequency band to the instantaneous frequency position corresponding to the modulus maximum value; Step S5: extracting the instantaneous frequency curve by using the local modulus maximum method; Step S3 is specifically: A local sliding window with a window width of is set in the frequency direction, and then the position of the maximum point of the modulus value in the local sliding window at each time is searched, the time-frequency coefficients in the window are reserved to construct a frequency band, and the coefficient values outside the frequency band are zeroed, as shown in equation (9). ​ (9) wherein are time-frequency coefficients of a generalized S-transform, is a parameter for adjusting the position of the Gaussian window in the time axis, and f is the frequency. Then, the phase is preserved according to equation (10) The corresponding instantaneous frequency The remaining positions are zeroed. (10)； wherein is the instantaneous frequency; The process of step S4 is shown in formula (11), the reserved time-frequency coefficients are rearranged in the frequency direction to obtain a clear time-frequency map of the local maximum synchronous squeezed generalized S transform ; (11)； wherein is a Kronig function; The implementation process of step S5 is as follows: First, search the frequency range at each time step. Find the maximum value of the modulus and record the position. Its expression is: (12) The resulting positions are then converted to corresponding instantaneous frequencies according to the following equation and connected together in time series to obtain an instantaneous frequency curve ; (13) In the formula, q is the maximum value of the frequency axis of the time-frequency graph, and r is the dimension of the local maximum synchronous squeezing generalized S transform coefficient matrix.

2. The instantaneous frequency identification method based on the local maximum synchronous squeezing generalized S transform algorithm according to claim 1, characterized in that: Step S1 is specifically: The generalized S transform is performed on the signal, and the expression is as follows: (1) where t is time, is a phase correction factor, i is the imaginary unit; is a Gaussian window function, which is expressed as: (2) In the formula, m and n are parameters for adjusting the time duration and decay speed of the Gaussian window. Accordingly, The frequency domain expression of is: (3) wherein is the Fourier transform of the signal whose expression is: (4) Wherein, α is a parameter for adjusting the width of the sliding window in the frequency direction.

3. The instantaneous frequency identification method based on the local maximum synchronous squeezing generalized S transform algorithm according to claim 2, characterized in that: Step S2 is specifically: Since any signal can be approximated by a sum of simple harmonics, consider a harmonic signal and take its Fourier transform: (5) Up to this point, Rewritten as: (6) For the partial derivatives with respect to are obtained: (7) At this time, the instantaneous frequency is expressed as: (8)。

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