A high-resolution time-frequency analysis method based on fractional wavelet transform

Abstract

The application provides a high-resolution time-frequency analysis method based on fractional wavelet transform, which comprises the following steps: calculating the fractional wavelet transform of a signal at an angle corresponding to a fractional Fourier transform domain where the energy of the signal is best gathered, so as to obtain a signal representation based on the fractional wavelet transform and combined with time and fractional scale; then, obtaining a signal representation combined with time and fractional frequency through the internal relation between the fractional scale and the fractional frequency; finally, obtaining a signal representation combined with time and frequency based on the fractional wavelet transform according to the relation between the fractional frequency and the frequency. Compared with the existing time-frequency analysis method based on the traditional wavelet transform, the time-frequency analysis method based on the fractional wavelet transform can further improve the resolution of time-frequency analysis.

Description

Technical Field

[0001] This invention belongs to the field of signal and information processing technology, and particularly relates to a high-resolution time-frequency analysis method based on fractional wavelet transform. Background Technology

[0002] Time-frequency analysis can characterize the evolution of a signal's spectrum over time and is an effective tool for processing non-stationary signals. Wavelet transform, as one of the most basic and commonly used time-frequency analysis methods, has been widely applied in both scientific research and engineering applications. However, wavelet transform essentially corresponds to a set of multi-scale frequency domain filters, suitable only for processing signals with optimal frequency domain energy concentration. For signals with non-optimal frequency domain energy concentration, the processing results are not optimal. For example, linear frequency modulated signals widely used in radar and communications are typical examples of signals with non-optimal frequency domain energy concentration. Therefore, a series of novel time-frequency analysis methods have emerged based on traditional wavelet transform. Among them, fractional wavelet transform, as a generalized form of traditional wavelet transform, has received considerable attention in recent years. However, fractional wavelet transform provides a joint representation of time and fractional scales, not a joint representation of time and frequency, and cannot directly characterize the time-varying characteristics of the spectrum of non-stationary signals. Therefore, this invention proposes a high-resolution time-frequency analysis method based on fractional wavelet transform. Summary of the Invention

[0003] The purpose of this invention is to solve the time-frequency analysis problem of non-stationary signals with non-optimal energy concentration in the frequency domain, and to propose a high-resolution time-frequency analysis method based on fractional wavelet transform.

[0004] This invention is achieved through the following technical solution: This invention proposes a high-resolution time-frequency analysis method based on fractional wavelet transform, the method comprising the following steps:

[0005] Step 1: Given the signal to be analyzed, i.e., the finite-energy signal f(t), calculate its fractional Fourier transform F. α (u), where the angle range is α∈(0,2π];

[0006] Step 2: Determine the optimal angle α in the fractional Fourier transform domain for the optimal energy concentration of the finite-energy signal f(t). opt ,Right now

[0007]

[0008] Step 3: Select the mother wavelet function ψ(t), whose Fourier transform Ψ(ω) satisfies Calculate the optimal energy concentration of a finite-energy signal f(t) in the fractional-order Fourier transform domain corresponding to the angle α. opt The fractional wavelet transform below, i.e.

[0009]

[0010] Step 4: Calculate the spectral center of the Fourier transform Ψ(ω) of the mother wavelet function ψ(t), i.e.

[0011]

[0012] Step 5: Based on the relationship between fractional scale and fractional frequency, i.e., a = E Ψ sinα / u, derived from the fractional wavelet transform coefficients in step three. Calculate the fractional wavelet transform coefficients represented by the joint time t and fractional frequency u. Right now

[0013]

[0014] Step Six: Based on the relationship between frequency ω and fractional frequency u, i.e., ω = ucscα - tcotα, the fractional wavelet transform coefficients expressed by the joint time t and fractional frequency u in Step Five are... Calculate the fractional wavelet transform coefficients represented by the joint time t and fractional frequency ω. Right now

[0015]

[0016] Furthermore, the method also includes the process of recovering the original signal from the result of time-frequency analysis processing, specifically as follows:

[0017] Step 7: Use the fractional wavelet transform coefficients represented by the joint time t and fractional frequency ω processed in Step 6. Based on the relationship between frequency ω and fractional frequency u, i.e., ω = ucscα - tcotα, the fractional wavelet transform coefficients represented by the joint time t and fractional frequency u are calculated. Right now

[0018]

[0019] Step 8: Use the joint time t and fractional frequency u obtained in Step 7 to represent the fractional wavelet transform coefficients. Based on the relationship between fractional scale and fractional frequency, i.e., a = E Ψ sinα / u, calculate the fractional wavelet transform coefficients represented by the joint time t and the fractional scale a. Right now

[0020]

[0021] Step 9: Use the joint time t and fractional scale a obtained in Step 8 to represent the fractional wavelet transform coefficients. By combining the inverse fractional wavelet transform, the original signal after time-frequency analysis can be recovered.

[0022]

[0023] The beneficial effects of this invention are:

[0024] The method described in this invention calculates the fractional wavelet transform of a signal at the angle corresponding to the optimal concentration of signal energy in the fractional Fourier transform domain. This concentrates the signal energy on a small number of fractional wavelet transform coefficients, which is beneficial for achieving sparse signal representation and improving algorithm efficiency. Furthermore, compared with existing time-frequency analysis methods based on traditional wavelet transforms, the time-frequency analysis method based on the optimal angle fractional wavelet transform can further improve the resolution of time-frequency analysis. Attached Figure Description

[0025] Figure 1 This is a flowchart of a time-frequency analysis method based on fractional wavelet transform.

[0026] Figure 2 The flowchart shows the process of recovering the original signal based on the time-frequency analysis results of fractional wavelet transform.

[0027] Figure 3 This is a schematic diagram of the results based on traditional wavelet transform time-frequency analysis.

[0028] Figure 4 This is a schematic diagram of the results of time-frequency analysis based on fractional wavelet transform. Detailed Implementation

[0029] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0030] For ease of analysis, we first introduce the definition of the fractional Fourier transform. For any finite-energy signal f(t) ∈ L... 2 The fractional Fourier transform of (R) is defined as

[0031]

[0032] In the formula, Represents the fractional Fourier transform operator, kernel function The expression is

[0033]

[0034] In the formula, k∈Z, α represents the angle of the fractional Fourier transform, the variable u is usually called the fractional frequency, and the coordinate axis in which it lies is usually called the fractional Fourier transform domain. Correspondingly, the formula for the inverse fractional Fourier transform is:

[0035]

[0036] In the formula, the superscript symbol * indicates the conjugate operation. In particular, when α = π / 2, the fractional Fourier transform degenerates into the traditional Fourier transform.

[0037] Furthermore, to simplify the analysis, the definition of the fractional-order Wigner-Ville distribution needs to be introduced. The definition of the fractional-order Wigner-Ville distribution of any finite-energy signal f(t) is as follows:

[0038]

[0039] In particular, when α = π / 2, the fractional Wigner-Ville distribution degenerates into the classical Wigner-Ville distribution, i.e.

[0040]

[0041] It can be seen that the fractional-order Wigner-Ville distribution provides a representation of the joint time *t* and fractional-order frequency *u*, while the classical Wigner-Ville distribution provides a representation of the joint time *t* and frequency *ω*. Comparing their definitions, we can further derive the intrinsic connection between the fractional-order Wigner-Ville distribution and the classical Wigner-Ville distribution, namely…

[0042]

[0043] Therefore, the relationship between frequency ω and fractional frequency u can be obtained, that is...

[0044] ω=ucscα-tcotα (7)

[0045] Furthermore, we introduce the definition of fractional wavelet transform. For any finite-energy signal f(t)∈L 2 The fractional wavelet transform of (R) is defined as

[0046]

[0047] In the formula, the kernel function ψ α,a,b The expression for (t) is

[0048]

[0049] In the formula, the fractional scale parameter a and the time shift parameter t satisfy: a∈R + , t∈R. Correspondingly, the inverse transform formula for the fractional wavelet transform is:

[0050]

[0051] In the formula, the constant C ψ satisfy

[0052]

[0053] In the formula, Ψ(ω) represents the Fourier transform of the mother wavelet function ψ(t). Furthermore, the definition of the fractional wavelet transform can be expressed in the form of the fractional Fourier transform domain, i.e.

[0054]

[0055] In the formula, Ψ(ucscα) represents the Fourier transform of the mother wavelet function ψ(t) (the transform elements are scaled by cscα). It can be seen that the fractional wavelet transform essentially corresponds to a set of multi-scale filters in the fractional Fourier transform domain, suitable for processing signals with optimal energy concentration in the fractional Fourier transform domain (a special case where the frequency domain is at angle α = π / 2). Linear frequency modulated signals widely used in radar, communication, and other electronic information systems are typical examples of signals with non-optimal frequency domain concentration, while signals with optimal energy concentration in the fractional Fourier transform domain are...

[0056] However, it should be noted that the fractional wavelet transform provides a representation of the joint time *t* and fractional scale *a*, not the desired representation of the joint time *t* and frequency *ω*. To address this issue, we first transform the representation of the joint time *t* and fractional scale *a* provided by the fractional wavelet transform into a representation of the joint time *t* and fractional frequency *u*. According to the theory of fractional wavelet transform, the fractional scale and fractional frequency have the following relationship:

[0057]

[0058] In the formula, E Ψ The spectral center of the Fourier transform Ψ(ω) of the mother wavelet function ψ(t) is represented by...

[0059]

[0060] Therefore, the definition of the fractional wavelet transform can be rewritten as a representation of the joint time t and fractional frequency u, i.e.

[0061]

[0062] Therefore, combined with equation (7), the definition of fractional wavelet transform can be rewritten as a representation combining time t and frequency ω, i.e.

[0063]

[0064] Based on the above analysis, the following describes a high-resolution time-frequency analysis method based on fractional wavelet transform proposed in this invention.

[0065] This invention proposes a high-resolution time-frequency analysis method based on fractional wavelet transform, the method comprising the following steps:

[0066] Step 1: Given the signal to be analyzed, i.e., the finite-energy signal f(t), calculate its fractional Fourier transform F. α (u), where the angle range is α∈(0,2π];

[0067] Step 2: Determine the optimal angle α in the fractional Fourier transform domain for the optimal energy concentration of the finite-energy signal f(t). opt ,Right now

[0068]

[0069] Step 3: Select the mother wavelet function ψ(t), whose Fourier transform Ψ(ω) satisfies Calculate the optimal energy concentration of a finite-energy signal f(t) in the fractional-order Fourier transform domain corresponding to the angle α. opt The fractional wavelet transform below, i.e.

[0070]

[0071] Step 4: Calculate the spectral center of the Fourier transform Ψ(ω) of the mother wavelet function ψ(t), i.e.

[0072]

[0073] Step 5: Based on the relationship between fractional scale and fractional frequency, i.e., a = E Ψ sinα / u, derived from the fractional wavelet transform coefficients in step three. Calculate the fractional wavelet transform coefficients represented by the joint time t and fractional frequency u. Right now

[0074]

[0075] Step Six: Based on the relationship between frequency ω and fractional frequency u, i.e., ω = ucscα - tcotα, the fractional wavelet transform coefficients expressed by the joint time t and fractional frequency u in Step Five are... Calculate the fractional wavelet transform coefficients represented by the joint time t and fractional frequency ω. Right now

[0076]

[0077] The above steps illustrate the time-frequency analysis process based on fractional wavelet transform. The method also includes the process of recovering the original signal from the time-frequency analysis result, specifically:

[0078] Step 7: Use the fractional wavelet transform coefficients represented by the joint time t and fractional frequency ω processed in Step 6. Based on the relationship between frequency ω and fractional frequency u, i.e., ω = ucscα - tcotα, the fractional wavelet transform coefficients represented by the joint time t and fractional frequency u are calculated. Right now

[0079]

[0080] Step 8: Use the joint time t and fractional frequency u obtained in Step 7 to represent the fractional wavelet transform coefficients. Based on the relationship between fractional scale and fractional frequency, i.e., a = E Ψ sinα / u, calculate the fractional wavelet transform coefficients represented by the joint time t and the fractional scale a. Right now

[0081]

[0082] Step 9: Use the joint time t and fractional scale a obtained in Step 8 to represent the fractional wavelet transform coefficients. By combining the inverse fractional wavelet transform, the original signal after time-frequency analysis can be recovered.

[0083]

[0084] The effects of this invention can be further illustrated by the following simulations:

[0085] Simulated signals It can be seen that the simulated signal f(t) contains three signal components, namely and For the simulated signal f(t), Figure 3 and Figure 4 Time-frequency analysis results based on traditional wavelet transform and fractional wavelet transform are presented respectively. It can be seen that, compared with the time-frequency analysis results based on traditional wavelet transform, the time-frequency analysis based on fractional wavelet transform can effectively show that the signal f(t) contains three signal components.

Claims

1. A high-resolution time-frequency analysis method based on fractional wavelet transform, characterized in that, The method includes the following steps: Step 1: Given the signal to be analyzed, i.e., the finite-energy signal f(t), calculate its fractional Fourier transform F. α (u), where the angle range is α∈(0,2π]; Step 2: Determine the optimal angle α in the fractional Fourier transform domain for the optimal energy concentration of the finite-energy signal f(t). opt ,Right now Step 3: Select the mother wavelet function ψ(t), whose Fourier transform Ψ(ω) satisfies Calculate the optimal energy concentration of a finite-energy signal f(t) in the fractional-order Fourier transform domain corresponding to the angle α. opt The fractional wavelet transform below, i.e. Step 4: Calculate the spectral center of the Fourier transform Ψ(ω) of the mother wavelet function ψ(t), i.e. Step 5: Based on the relationship between fractional scale and fractional frequency, i.e., a = E Ψ sinα / u, derived from the fractional wavelet transform coefficients in step three. Calculate the fractional wavelet transform coefficients represented by the joint time t and fractional frequency u. Right now Step Six: Based on the relationship between frequency ω and fractional frequency u, i.e., ω = ucscα - tcotα, the fractional wavelet transform coefficients expressed by the joint time t and fractional frequency u in Step Five are... Calculate the fractional wavelet transform coefficients represented by the joint time t and fractional frequency ω. Right now 2. The method according to claim 1, characterized in that, The method also includes a process for recovering the original signal from the result of time-frequency analysis processing, specifically: Step 7: Use the fractional wavelet transform coefficients represented by the joint time t and fractional frequency ω processed in Step 6. Based on the relationship between frequency ω and fractional frequency u, i.e., ω = ucscα - tcotα, the fractional wavelet transform coefficients represented by the joint time t and fractional frequency u are calculated. Right now Step 8: Use the joint time t and fractional frequency u obtained in Step 7 to represent the fractional wavelet transform coefficients. Based on the relationship between fractional scale and fractional frequency, i.e., a = E Ψ sinα / u, calculate the fractional wavelet transform coefficients represented by the joint time t and the fractional scale a. Right now Step 9: Use the joint time t and fractional scale a obtained in Step 8 to represent the fractional wavelet transform coefficients. By combining the inverse fractional wavelet transform, the original signal after time-frequency analysis can be recovered.

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