A high-resolution time-frequency rearrangement distribution method in fractional wavelet transform domain

Abstract

The application provides a high-resolution time-frequency rearrangement distribution method in a fractional wavelet transform domain, which calculates fractional wavelet transforms of a signal under three different mother wavelets at an angle corresponding to a fractional Fourier transform domain where signal energy is optimally gathered, and then obtains an estimation of signal instantaneous frequency and group delay. Then, the energy of the signal is rearranged to time-frequency points determined by the instantaneous frequency and the group delay on a time-frequency plane, so as to obtain a high-resolution signal time-frequency representation based on the fractional wavelet transform. Compared with a time-frequency rearrangement distribution based on a traditional wavelet transform, the time-frequency rearrangement distribution based on the fractional wavelet transform can further improve the resolution of signal time-frequency analysis through the selection of a free parameter alpha, so that the time-frequency structure characteristics of the signal can be clearly displayed on the time-frequency plane.

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 rearrangement distribution method in the fractional wavelet transform domain. Background Technology

[0002] In engineering applications such as radar, communication, medicine, and mechanical vibration, actual physical signals are often non-stationary, meaning their frequencies vary over time. Time-frequency analysis is a fundamental tool for analyzing non-stationary signals. Among these, wavelet transform is the most common time-frequency analysis method, capable of obtaining multi-resolution analysis results for signals at different scales. However, the time-frequency resolution of wavelet transform depends on frequency, exhibiting characteristics of low resolution at high frequencies and high resolution at low frequencies. Furthermore, time-frequency resolution is limited by the uncertainty principle, creating a contradiction of mutual constraints. Therefore, time-frequency rearrangement distribution methods based on wavelet transform have emerged. The prerequisite for this method is that there is no aliasing among the signal components on the time-frequency plane determined by the wavelet transform. Especially when the signal components are very similar on the time-frequency plane, extremely high time-frequency resolution is required from the wavelet transform. However, wavelet transform can only adjust the time-frequency resolution horizontally and vertically on the time-frequency plane through time translation and scale scaling, failing to fully utilize the signal's inherent time-frequency geometric characteristics. Therefore, this invention proposes a high-resolution time-frequency rearrangement distribution method based on fractional wavelet transform. Summary of the Invention

[0003] The purpose of this invention is to solve the problems in the prior art and to propose a high-resolution time-frequency rearrangement distribution method in the fractional wavelet transform domain.

[0004] This invention is achieved through the following technical solution: This invention proposes a high-resolution time-frequency rearrangement distribution method in the fractional wavelet transform domain, the method comprising the following steps:

[0005] Step 1: Given the signal to be analyzed, i.e., the linear frequency modulated signal f(t), calculate its fractional Fourier transform F. α (u), where the angle ranges from α∈(0,2π];

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

[0007] Step 3: Select the mother wavelet function ψ(t), whose Fourier transform Ψ(ω) satisfies The optimal energy concentration of a finite-energy signal, i.e., a linear frequency modulated signal f(t), is calculated in the fractional Fourier transform domain corresponding to the angle α. opt The fractional wavelet transform below, i.e.

[0008]

[0009] Step 4: Calculate the mother wavelet function Tψ(t) = t·ψ(t), and calculate the angle α corresponding to the optimal energy concentration fractional Fourier transform domain of the linear frequency modulated signal f(t). opt The fractional wavelet transform below, i.e.

[0010]

[0011] Step 5: Calculate the mother wavelet function And calculate the angle α corresponding to the optimal energy concentration fractional Fourier transform domain of the linear frequency modulated signal f(t). opt The fractional wavelet transform below, i.e.

[0012]

[0013] Step 6: Calculate the optimal energy focusing angle α of the linear frequency modulated signal f(t). opt The estimation of the next instantaneous frequency, i.e.

[0014]

[0015] Step 7: Calculate the optimal energy focusing angle α of the linear frequency modulated signal f(t). opt The estimation of the group delay, i.e.

[0016]

[0017] Step 8: Calculate the optimal energy focusing angle α of the linear frequency modulated signal f(t). opt The time-frequency rearrangement distribution of the lower fractional wavelet transform, i.e.

[0018]

[0019] In the formula,

[0020] The beneficial effects of this invention are:

[0021] This invention proposes a high-resolution time-frequency rearrangement distribution method in the fractional wavelet transform domain. The method calculates the fractional wavelet transform of the signal under three different mother wavelets at the angle corresponding to the optimal concentration of signal energy in the fractional Fourier transform domain, thereby obtaining estimates of the signal's instantaneous frequency and group delay. Then, the signal energy is rearranged to the time-frequency points determined by the instantaneous frequency and group delay on the time-frequency plane, resulting in a high-resolution time-frequency representation of the signal based on the fractional wavelet transform. Compared with time-frequency rearrangement distribution based on traditional wavelet transform, the time-frequency rearrangement distribution based on the fractional wavelet transform can further improve the resolution of signal time-frequency analysis through the selection of the free parameter α, thus clearly displaying the time-frequency structural characteristics of the signal on the time-frequency plane. Attached Figure Description

[0022] Figure 1 This is a block diagram illustrating the principle of a high-resolution time-frequency rearrangement distribution method in the fractional wavelet transform domain.

[0023] Figure 2 It is a theoretical curve showing how the frequency of the simulated signal changes over time.

[0024] Figure 3 This is a schematic diagram of the time-frequency rearrangement result based on the existing wavelet transform.

[0025] Figure 4 This is a schematic diagram of the time-frequency rearrangement result based on fractional wavelet transform. Detailed Implementation

[0026] 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.

[0027] For ease of analysis, we first introduce the concept of fractional Fourier transform. For any finite-energy signal f(t)∈L... 2 (R), the fractional Fourier transform is defined as

[0028]

[0029] In the formula, the kernel function The expression is

[0030]

[0031] 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:

[0032]

[0033] In the formula, the superscript symbol * This represents the conjugate operation. Specifically, when α = π / 2, the fractional Fourier transform degenerates into the traditional Fourier transform. Furthermore, the definition of the fractional wavelet transform is introduced. For any finite-energy signal f(t) ∈ L... 2 (R), the fractional wavelet transform is defined as

[0034]

[0035] In the formula, the kernel function ψ α,a,b The expression for (t) is:

[0036]

[0037] In the formula, the fractional scale parameter a and the time shift parameter t satisfy: a∈R + , t∈R. Furthermore, the definition of the fractional wavelet transform can be expressed in the form of the fractional Fourier transform domain, i.e.

[0038]

[0039] 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...

[0040] When the angle α = π / 2, the fractional wavelet transform degenerates into a wavelet transform, i.e.

[0041]

[0042] Furthermore, the relationship between fractional wavelet transform and wavelet transform can be obtained, i.e.

[0043]

[0044] In the formula

[0045]

[0046] To estimate the instantaneous frequency and group delay of the time-frequency rearrangement distribution, it is first necessary to derive the intrinsic relationship between wavelet transform and the Wegener-Weil distribution. For this purpose, the concept of the Wegener-Weil distribution needs to be introduced. For any finite-energy signal f(t), the Wegener-Weil distribution is defined as follows:

[0047]

[0048] According to time-frequency analysis theory, the signal The wavelet transform and the Wegener-Weil distribution have the following intrinsic connection:

[0049]

[0050] At the same time, signals can be obtained. instantaneous frequency Group delay The estimate is

[0051]

[0052]

[0053] Based on the relationship between wavelet transform and Wegener-Weil distribution, we can further obtain...

[0054]

[0055]

[0056] In the formula, Indicates taking the real part, This indicates taking the imaginary part, Tψ(t) = tψ(t), and Substituting equation (8) into equations (14) and (15), we can obtain the signal based on fractional wavelet transform. instantaneous frequency Group delay The estimate, i.e.

[0057]

[0058]

[0059] In practice, it is necessary to obtain the instantaneous frequency of the signal f(t). Group delay The estimation. Equation (9) shows the following relationship between them.

[0060]

[0061]

[0062] Therefore, the instantaneous frequency of the signal f(t) at angle α is estimated. Estimation of group delay for

[0063]

[0064]

[0065] Based on this, the fractional wavelet transform time-frequency rearrangement distribution is defined as follows:

[0066]

[0067] In the formula,

[0068] In particular, when α = π / 2, the fractional wavelet transform time-frequency rearrangement distribution degenerates into the traditional wavelet transform time-frequency rearrangement distribution.

[0069] Specifically, this invention proposes a high-resolution time-frequency rearrangement distribution method in the fractional wavelet transform domain, the method comprising the following steps:

[0070] Step 1: Given the signal to be analyzed, i.e., the linear frequency modulated signal f(t), calculate its fractional Fourier transform F. α (u), where the angle ranges from α∈(0,2π];

[0071] Step 2: Determine the optimal angle α for the optimal concentration of signal f(t) energy in the fractional Fourier transform domain. opt ,Right now

[0072] Step 3: Select the mother wavelet function ψ(t), whose Fourier transform Ψ(ω) satisfies The optimal energy concentration of a finite-energy signal, i.e., a linear frequency modulated signal f(t), is calculated in the fractional Fourier transform domain corresponding to the angle α. opt The fractional wavelet transform below, i.e.

[0073]

[0074] Step 4: Calculate the mother wavelet function Tψ(t) = t·ψ(t), and calculate the angle α corresponding to the optimal energy concentration fractional Fourier transform domain of the linear frequency modulated signal f(t). opt The fractional wavelet transform below, i.e.

[0075]

[0076] Step 5: Calculate the mother wavelet function And calculate the angle α corresponding to the optimal energy concentration fractional Fourier transform domain of the linear frequency modulated signal f(t). optThe fractional wavelet transform below, i.e.

[0077]

[0078] Step 6: Calculate the optimal energy focusing angle α of the linear frequency modulated signal f(t). opt The estimation of the next instantaneous frequency, i.e.

[0079]

[0080] Step 7: Calculate the optimal energy focusing angle α of the linear frequency modulated signal f(t). opt The estimation of the group delay, i.e.

[0081]

[0082] Step 8: Calculate the optimal energy focusing angle α of the linear frequency modulated signal f(t). opt The time-frequency rearrangement distribution of the lower fractional wavelet transform, i.e.

[0083]

[0084] In the formula,

[0085] The effects of this invention can be further illustrated by the following simulations:

[0086] The expression for the simulated signal is:

[0087]

[0088] It can be seen that the simulated signal contains four signal components, namely and Figure 2 The theoretical curve of its frequency changing with time is given based on the analytical expression of the signal. Figure 3 and Figure 4 The results of time-frequency rearrangement based on existing wavelet transform and fractional wavelet transform are presented respectively when the mother wavelet function is Morlet wavelet. It can be seen that, compared with wavelet transform, the time-frequency rearrangement result obtained by fractional wavelet transform can clearly show the component components contained in signal f(t), which can further improve the resolution of time-frequency analysis.

Claims

1. A high-resolution time-frequency rearrangement distribution method in the fractional wavelet transform domain, characterized in that, The method includes the following steps: Step 1: Given the signal to be analyzed, i.e., the linear frequency modulated signal f(t), calculate its fractional Fourier transform F. α (u), where the angle ranges from α∈(0,2π]; Step 2: Determine the optimal angle α for the optimal concentration of signal f(t) energy in the fractional Fourier transform domain. opt ,Right now Step 3: Select the mother wavelet function ψ(t), whose Fourier transform Ψ(ω) satisfies The optimal energy concentration of a finite-energy signal, i.e., a linear frequency modulated signal f(t), is calculated in the fractional Fourier transform domain corresponding to the angle α. opt The fractional wavelet transform below, i.e. Step 4: Calculate the mother wavelet function Tψ(t) = t·ψ(t), and calculate the angle α corresponding to the optimal energy concentration fractional Fourier transform domain of the linear frequency modulated signal f(t). opt The fractional wavelet transform below, i.e. Step 5: Calculate the mother wavelet function And calculate the angle α corresponding to the optimal energy concentration fractional Fourier transform domain of the linear frequency modulated signal f(t). opt The fractional wavelet transform below, i.e. Step 6: Calculate the optimal energy focusing angle α of the linear frequency modulated signal f(t). opt The estimation of the next instantaneous frequency, i.e. Step 7: Calculate the optimal energy focusing angle α of the linear frequency modulated signal f(t). opt The estimation of the group delay, i.e. Step 8: Calculate the optimal energy focusing angle α of the linear frequency modulated signal f(t). opt The time-frequency rearrangement distribution of the lower fractional wavelet transform, i.e. In the formula,

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