CN117113003A - High-resolution time-frequency analysis method based on fractional order wavelet transformation - Google Patents

High-resolution time-frequency analysis method based on fractional order wavelet transformation Download PDF

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CN117113003A
CN117113003A CN202311070813.3A CN202311070813A CN117113003A CN 117113003 A CN117113003 A CN 117113003A CN 202311070813 A CN202311070813 A CN 202311070813A CN 117113003 A CN117113003 A CN 117113003A
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fractional
frequency
wavelet transformation
time
fractional order
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刘晓萍
史军
门子俊
孙德华
陶然
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Beijing Institute of Technology BIT
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Abstract

The invention provides a high-resolution time-frequency analysis method based on fractional wavelet transformation, which calculates the fractional wavelet transformation of a signal under the angle corresponding to a fractional Fourier transformation domain with optimal aggregation of signal energy, so as to obtain the signal representation based on joint time and fractional scale of the fractional wavelet transformation. Then, by the internal relation between the fractional scale and the fractional frequency, a signal representation of the joint time and the fractional frequency is obtained. Finally, according to the relation between the fractional order frequency and the frequency, the signal representation of the joint time and the frequency based on fractional order wavelet transformation is obtained. Compared with the traditional time-frequency analysis method based on the wavelet transformation, the time-frequency analysis method based on the fractional order wavelet transformation can further improve the resolution of time-frequency analysis.

Description

High-resolution time-frequency analysis method based on fractional order wavelet transformation
Technical Field
The invention belongs to the technical field of signal and information processing, and particularly relates to a high-resolution time-frequency analysis method based on fractional order wavelet transformation.
Background
Time-frequency analysis can characterize the evolution of the signal spectrum over time, an effective means of non-stationary signal processing. Wavelet transformation is used as a most basic and most commonly used time-frequency analysis method, and is widely applied in the fields of scientific research and engineering application. However, wavelet transforms essentially correspond to a set of multi-scale frequency domain filters, which are only suitable for processing signals with optimal aggregation of frequency domain energy. The result of the processing is not optimal for signals with non-optimal aggregation of frequency domain energy. For example, chirp signals, which are widely used in radar, communication, and the like, are typical frequency domain energy non-optimal aggregate signals. In view of this, a series of novel time-frequency analysis methods emerge on the basis of the conventional wavelet transform. Among them, fractional order wavelet transform has been attracting attention in recent years as a generalized form of conventional wavelet transform. However, fractional wavelet transforms provide a joint representation of time and fractional scales, not a joint time and frequency representation, and do not directly characterize the time-varying characteristics of the non-stationary signal spectrum. Therefore, the invention provides a high-resolution time-frequency analysis method based on fractional order wavelet transformation.
Disclosure of Invention
The invention aims to solve the problem of time-frequency analysis of non-stationary signals with non-optimal aggregation of frequency domain energy, and provides a high-resolution time-frequency analysis method based on fractional order wavelet transformation.
The invention is realized by the following technical scheme, and provides a high-resolution time-frequency analysis method based on fractional order wavelet transformation, which comprises the following steps:
step one, giving a signal to be analyzed, namely an energy limited signal F (t), and calculating fractional Fourier transform F of the signal α (u) wherein the angular range is alpha E (0, 2 pi)]；
Step two, determining the optimal angle alpha of the energy-limited signal f (t) energy optimal aggregation fractional Fourier transform domain opt I.e.
Step three, selecting a mother wavelet function psi (t), wherein the Fourier transform psi (omega) meets the following conditionCalculating the corresponding angle alpha of the energy-limited signal f (t) energy optimal aggregation fractional Fourier transform domain opt Lower fractional order wavelet transform, i.e
Step four, calculating the spectrum center of the Fourier transform ψ (omega) of the mother wavelet function ψ (t), namely
Step five, according to the relation between the fractional scale and the fractional frequency, namely a=E Ψ sin alpha/u, which is obtained by fractional order wavelet transformation coefficient in step threeCalculating fractional wavelet transformation coefficient represented by joint time t and fractional frequency u>I.e.
Step six, according to the relation between the frequency omega and the fractional frequency u, namely omega=ucsc alpha-tcotalpha, the fractional wavelet transformation coefficient represented by the joint time t and the fractional frequency u in the step fiveCalculating fractional wavelet transformation coefficient ++represented by joint time t and fractional frequency ω>I.e.
Further, the method further comprises a process of recovering the original signal from the result after the time-frequency analysis processing, specifically:
step seven, using the joint time t and fractional order frequency omega after processing in step six to represent fractional order wavelet transformation coefficientCalculating a fractional wavelet transformation coefficient ++represented by the joint time t and the fractional frequency u according to the relation between the frequency ω and the fractional frequency u, namely ω=ucsc α -tcotα>I.e.
Step eight, utilizing the joint time t and fractional order frequency u obtained in step seven to represent fractional order wavelet transformation coefficientBased on the relationship between fractional scale and fractional frequency, i.e. a=e Ψ sin alpha/u, calculating a fractional wavelet transformation coefficient +.>I.e.
Step nine, utilizing the combination obtained in step eightFractional wavelet transform coefficients expressed by time t and fractional scale aThe original signal after time-frequency analysis processing can be recovered by combining the inverse transformation of fractional order wavelet transformation, namely
The invention has the beneficial effects that:
according to the method, the fractional wavelet transform of the signal is calculated under the angle corresponding to the fractional Fourier transform domain of the optimal aggregation of the signal energy, so that the signal energy can be concentrated on a few fractional wavelet transform coefficients, sparse representation of the signal is facilitated, and the algorithm operation efficiency can be improved. In addition, compared with the traditional time-frequency analysis method based on the traditional wavelet transformation, the time-frequency analysis method based on the optimal angle fractional order wavelet transformation can further improve the resolution of time-frequency analysis.
Drawings
Fig. 1 is a block flow diagram of a time-frequency analysis method based on fractional order wavelet transform.
Fig. 2 is a block flow diagram of recovering an original signal based on the fractional wavelet transform time-frequency analysis result.
Fig. 3 is a schematic diagram of the result of time-frequency analysis based on conventional wavelet transform.
Fig. 4 is a schematic diagram of the results of fractional order wavelet transform based time-frequency analysis.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
For the convenience ofFor analysis, a definition of a fractional fourier transform is first introduced. Arbitrary energy limited signal f (t) ∈L 2 The fractional Fourier transform of (R) is defined as
In the method, in the process of the invention,representing a fractional order Fourier transform operator, kernel function->The expression of (C) is as follows
Where k e Z, α represents the angle of the fractional fourier transform, the variable u is usually called the fractional frequency, and the coordinate axis on which it is located is usually called the fractional fourier transform domain. Accordingly, the inverse transform of the fractional Fourier transform is formulated as
In the formula, superscript symbol indicates conjugate operation. In particular, when α=pi/2, the fractional fourier transform is degraded to a conventional fourier transform.
In addition, to simplify the analysis, a definition of fractional order Wigner-Ville distribution needs to be introduced. Definition of fractional order Wigner-Ville distribution of arbitrary energy limited signal f (t) is
In particular, when α=pi/2, the fractional order Wigner-Ville distribution is degraded to a classical Wigner-Ville distribution, i.e.
It can be seen that the fractional order Wigner-Ville distribution provides a representation of the joint time t and the fractional order frequency u, whereas the classical Wigner-Ville distribution provides a representation of the joint time t and the frequency ω. The definition of the two is compared, so that the inherent relation between the fractional order Wigner-Ville distribution and the classical Wigner-Ville distribution can be further obtained, namely
Thus, the relationship between the frequency ω and the fractional order frequency u can be obtained, i.e
ω＝ucscα-tcotα (7)
Further, a definition of fractional order wavelet transform is introduced. For any energy limited signal f (t) ∈L 2 The fractional order wavelet transform of (R) is defined as
In the formula, the kernel function psi α,a,b The expression of (t) is
Wherein, the fractional scale parameter a and the time shift parameter t satisfy the following conditions: a epsilon R + T is R. Accordingly, the inverse transformation formula of fractional order wavelet transform is
Wherein, constant C ψ Satisfy the following requirements
Where 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 a fractional fourier transform domain, i.e
Where ψ (ucsc α) represents the fourier transform of the mother wavelet function ψ (t) (the transform element scales cscα). It can be seen that the fractional wavelet transform essentially corresponds to a multi-scale filter of a set of fractional fourier transform domains, suitable for processing signals with the best energy concentration of the fractional fourier transform domain (the special case when the frequency domain is at its angle α=pi/2). The chirp signal widely used in electronic information systems such as radar and communication is a typical frequency domain non-optimal aggregate, and the signal with fractional fourier transform domain energy optimal aggregate.
However, it should be noted that fractional order wavelet transforms provide representations of joint time t and fractional order scale a, and not of the desired joint time t and frequency ω. To solve this problem, first, the representations of the joint time t and the fractional scale a provided by the fractional wavelet transform are converted into representations of the joint time t and the fractional frequency u. According to fractional wavelet transformation theory, the fractional scale and fractional frequency have the following relation
Wherein E is Ψ Representing the spectral center of the fourier transform ψ (ω) of the mother wavelet function ψ (t), i.e
The definition of the fractional wavelet transform can then be rewritten as a representation of the joint time t and the fractional frequency u, i.e
Whereby, in combination with equation (7), the fractional wavelet transform definition can be rewritten to a representation of the joint time t and frequency ω, i.e
Based on the above analysis, the high-resolution time-frequency analysis method based on fractional order wavelet transformation provided by the invention is described below.
The invention provides a high-resolution time-frequency analysis method based on fractional order wavelet transformation, which comprises the following steps:
step one, giving a signal to be analyzed, namely an energy limited signal F (t), and calculating fractional Fourier transform F of the signal α (u) wherein the angular range is alpha E (0, 2 pi)]；
Step two, determining the optimal angle alpha of the energy-limited signal f (t) energy optimal aggregation fractional Fourier transform domain opt I.e.
Step three, selecting a mother wavelet function psi (t), wherein the Fourier transform psi (omega) meets the following conditionCalculating the corresponding angle alpha of the energy-limited signal f (t) energy optimal aggregation fractional Fourier transform domain opt Lower fractional order wavelet transform, i.e
Step four, calculating the spectrum center of the Fourier transform ψ (omega) of the mother wavelet function ψ (t), namely
Step five, according to the relation between the fractional scale and the fractional frequency, namely a=E Ψ sin alpha/u, which is obtained by fractional order wavelet transformation coefficient in step threeCalculating fractional wavelet transformation coefficient represented by joint time t and fractional frequency u>I.e.
Step six, according to the relation between the frequency omega and the fractional frequency u, namely omega=ucsc alpha-tcotalpha, the fractional wavelet transformation coefficient represented by the joint time t and the fractional frequency u in the step fiveCalculating fractional wavelet transformation coefficient ++represented by joint time t and fractional frequency ω>I.e.
The steps above provide a time-frequency analysis process based on fractional order wavelet transformation, and the method further includes a process of recovering an original signal from a result after time-frequency analysis processing, specifically:
step seven, using the joint time t and fractional order frequency omega after processing in step six to represent fractional order wavelet transformation coefficientCalculating a fractional wavelet transformation coefficient ++represented by the joint time t and the fractional frequency u according to the relation between the frequency ω and the fractional frequency u, namely ω=ucsc α -tcotα>I.e.
Step eight, utilizing the joint time t and fractional order frequency u obtained in step seven to represent fractional order wavelet transformation coefficientBased on the relationship between fractional scale and fractional frequency, i.e. a=e Ψ sin alpha/u, calculating a fractional wavelet transformation coefficient +.>I.e.
Step nine, using the joint time t and the fractional order scale a obtained in step eight to represent the fractional order wavelet transformation coefficientThe original signal after time-frequency analysis processing can be recovered by combining the inverse transformation of fractional order wavelet transformation, namely
The effect of the invention can be further illustrated by the following simulations:
simulation signalIt can be seen that the artificial signal f (t) contains three signal components, namely +.>And->For the simulation signal f (t), fig. 3 and fig. 4 show the results of time-frequency analysis based on the conventional wavelet transform and the fractional wavelet transform, respectively. It can be seen that, compared with the time-frequency analysis result based on the conventional wavelet transform, the time-frequency analysis based on the fractional wavelet transform can effectively show that the signal f (t) contains three signal components.

Claims (2)

1. A high resolution time-frequency analysis method based on fractional order wavelet transform, characterized in that the method comprises the steps of:
step one, giving a signal to be analyzed, namely an energy limited signal F (t), and calculating fractional Fourier transform F of the signal α (u) wherein the angular range is alpha E (0, 2 pi)]；
Step two, determining the optimal angle alpha of the energy-limited signal f (t) energy optimal aggregation fractional Fourier transform domain opt I.e.
Step three, selecting a mother wavelet function psi (t), wherein the Fourier transform psi (omega) meets the following conditionCalculating the corresponding angle alpha of the energy-limited signal f (t) energy optimal aggregation fractional Fourier transform domain opt Lower fractional order wavelet transform, i.e
Step four, calculating the spectrum center of the Fourier transform ψ (omega) of the mother wavelet function ψ (t), namely
Step five, according to the relation between the fractional scale and the fractional frequency, namely a=E Ψ sin alpha/u, which is obtained by fractional order wavelet transformation coefficient in step threeCalculating fractional wavelet transform coefficient expressed by joint time t and fractional frequency uI.e.
Step six, according to the relation between the frequency omega and the fractional frequency u, namely omega=ucsc alpha-tcotalpha, the fractional wavelet transformation coefficient represented by the joint time t and the fractional frequency u in the step fiveCalculating fractional wavelet transformation coefficient ++represented by joint time t and fractional frequency ω>I.e.
2. The method according to claim 1, further comprising the step of recovering the original signal from the result of the time-frequency analysis process, specifically:
step seven, using the joint time t and fractional order frequency omega after processing in step six to represent fractional order wavelet transformation coefficientCalculating a fractional wavelet transformation coefficient ++represented by the joint time t and the fractional frequency u according to the relation between the frequency ω and the fractional frequency u, namely ω=ucsc α -tcotα>I.e.
Step eight, utilizing the joint time t and fractional order frequency u obtained in step seven to represent fractional order wavelet transformation coefficientBased on the relationship between fractional scale and fractional frequency, i.e. a=e Ψ sin alpha/u, calculating a fractional wavelet transformation coefficient +.>I.e.
Step nine, using the joint time t and the fractional order scale a obtained in step eight to represent the fractional order wavelet transformation coefficientThe original signal after time-frequency analysis processing can be recovered by combining the inverse transformation of fractional order wavelet transformation, namely
CN202311070813.3A 2023-08-24 2023-08-24 High-resolution time-frequency analysis method based on fractional order wavelet transformation Pending CN117113003A (en)

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