A data-driven fractional wavelet transform adaptive signal decomposition and reconstruction method

Abstract

The present invention proposes a data-driven fractional wavelet transform adaptive signal decomposition and reconstruction method. The method determines the support interval of the signal in the fractional Fourier transform domain where the signal energy is optimally concentrated, and constructs a data-driven fractional wavelet basis function for signal decomposition based on the fractional wavelet transform theory. Then, through a fractional convolution operation, the constructed fractional wavelet basis function is used as the convolution kernel to achieve filtering decomposition of the signal. Furthermore, a fractional wavelet basis function for signal reconstruction is designed based on the constructed fractional wavelet basis function, and a fractional convolution operation is used based on the signal decomposition result to achieve complete reconstruction of the signal. Compared with the existing method, it can avoid the problem that the signal energy diffusion in the frequency domain causes the components of the signal to overlap with each other and cannot be separated.

Description

Technical Field

[0001] The present invention belongs to the technical field of signal and information processing, and in particular relates to a data-driven fractional-order wavelet transform adaptive signal decomposition and reconstruction method. Background Art

[0002] Conventional signal analysis is mostly based on basis function expansions and requires the signal to be stationary. This lacks adaptability to complex changes in signal morphology, making it difficult to accurately extract the rich characteristic information contained in the signal. Applying appropriate signal analysis methods to effectively extract characteristic information from complex, non-stationary signals is a fundamental issue in signal processing.

[0003] Time-varying spectra are a typical characteristic of non-stationary signals. To effectively analyze complex non-stationary signals, a series of signal analysis methods based on the joint time-frequency domain have been proposed, such as the short-time Fourier transform, wavelet transform, and time-frequency distribution. However, most of these methods are based on integral transforms, requiring the manual design and selection of appropriate basis functions to match the signal's characteristic components based on signal characteristics. To overcome this problem, adaptive modal decomposition methods, represented by empirical mode decomposition (EMD), have emerged. EMD eliminates the need to design basis functions based on a priori knowledge; the decomposition is performed a posteriori, driven entirely by data, and exhibits strong signal adaptability. However, with increasing research and application, EMD has also exposed numerous shortcomings, such as a lack of rigorous mathematical support, modal aliasing, and end-point effects. In response to this, the empirical wavelet transform (EMT) was proposed within the framework of wavelet theory. The empirical wavelet transform has the advantages of both empirical mode decomposition and wavelet analysis. It adaptively segments the Fourier transform of the original signal, constructs a framework wavelet filter in each segmented interval, and obtains the modal component of the amplitude-frequency modulation signal with a compactly supported Fourier transform spectrum, thereby achieving adaptive decomposition of the signal. Compared with empirical mode decomposition, empirical wavelet has a solid mathematical foundation, can obtain stable signal decomposition, and has high computational efficiency. However, the wavelet filter is essentially a linear time-invariant filter in the Fourier transform domain, and is only suitable for stationary signals with optimal energy concentration in the Fourier transform domain. For those signals with non-optimal energy concentration in the Fourier transform domain, the processing results of the empirical wavelet transform are not optimal. In view of this, the present invention proposes a data-driven empirical fractional-order wavelet transform signal decomposition and reconstruction method. Summary of the Invention

[0004] The present invention aims to address the problem of adaptive decomposition and reconstruction of nonstationary signals with suboptimal frequency domain energy concentration. It provides a data-driven fractional wavelet transform adaptive signal decomposition and reconstruction method. By utilizing the fractional wavelet transform tool, this method designs data-driven fractional wavelet basis functions, enabling adaptive decomposition and complete reconstruction of the signal.

[0005] The present invention is implemented by the following technical solution. The present invention proposes a data-driven fractional wavelet transform adaptive signal decomposition and reconstruction method, which includes the following steps:

[0006] Step 1: Given the signal to be decomposed f(t), calculate the angle α corresponding to the fractional Fourier transform domain where the energy of the signal f(t) is optimally concentrated. opt ,Right now

[0007]

[0008] Step 2: Calculate the α where the signal to be decomposed f(t) is best concentrated in its energy opt The support interval U in the angle fractional Fourier transform domain n =[u n ,u n+1 ], where n m ≤n≤n M and n,n m ,n M ∈Z; at the same time, calculate the support interval center c n ;

[0009] Step 3: According to the specific type of the calculated support interval, calculate the center of the support interval and construct the fractional wavelet basis function for signal decomposition. The following three cases are explained:

[0010] (1) If the signal to be decomposed f(t) is at the α where its energy is best concentrated opt The support interval in the angle fractional Fourier transform domain belongs to the finite support interval, that is, Where -∞<n m ≤n≤n M <+∞ and n,n m ,n M ∈Z; then

[0011]

[0012] (2) If the signal to be decomposed f(t) is at the α where its energy is best concentrated opt The support interval on the angle fractional Fourier transform domain belongs to the infinite support interval on the left end, that is, where -∞=n m ≤n≤n M <+∞ and n,n m ,n M ∈Z, then the finite support interval The center of and its corresponding fractional wavelet basis function expressions are:

[0013]

[0014] However, for the infinite support interval on the left Then we have:

[0015]

[0016]

[0017] (3) If the signal to be decomposed f(t) is at the α where its energy is best concentrated opt The support interval on the angle fractional Fourier transform domain belongs to the infinite support interval on the right end Where -∞<n m ≤n≤n M =+∞ and n,n m ,n M ∈Z, then the finite support interval The center of and its corresponding fractional wavelet basis function expressions are:

[0018]

[0019] However, for the rightmost infinite support interval Then we have:

[0020]

[0021]

[0022] Step 4: Use the inverse transform of Fourier transform to obtain the time domain form of the fractional wavelet basis function for signal decomposition based on data drive, that is,

[0023]

[0024] Step 5: Solve the fractional-order wavelet basis function ψ for signal reconstruction based on the fractional-order wavelet basis function constructed for signal decomposition in the fractional-order Fourier transform domain. n (t), i.e.

[0025]

[0026] Step 6: Use the inverse transform of Fourier transform to obtain the time domain form of the fractional-order wavelet basis function used for signal reconstruction, that is:

[0027]

[0028] Step 7: Use the constructed fractional wavelet basis function ψ n (t), the signal f(t) is decomposed by fractional convolution operation, and the decomposed signal is:

[0029]

[0030] Step 8: Based on the decomposed signal Using the constructed fractional wavelet basis function φ n (t), the signal is completely reconstructed through fractional-order convolution operation, that is:

[0031]

[0032] The beneficial effects of the present invention are:

[0033] This paper proposes a data-driven fractional wavelet transform adaptive signal decomposition and reconstruction method. This method constructs data-driven fractional wavelet basis functions in the fractional Fourier transform domain, where signal energy is optimally concentrated, to achieve adaptive signal decomposition and complete signal reconstruction based on the decomposition results. Compared with existing methods, this method avoids the problem of signal energy diffusion in the frequency domain, which causes signal components to overlap and become unseparable. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 This is the principle block diagram of the data-driven fractional wavelet transform adaptive signal decomposition and reconstruction method.

[0035] Figure 2 It is the time domain waveform of the simulation signal f(t).

[0036] Figure 3 The time domain waveforms of the component signals f1(t), f2(t) and f3(t) of the simulation signal are as follows: Figure 3 (a) is the time domain waveform of the component signal f1(t), Figure 3 (b) is the time domain waveform of the component signal f2(t). Figure 3 (c) is the time domain waveform of the component signal f3(t).

[0037] Figure 4 is the spectrum and angle α of the simulated signal f(t) opt =-0.01 fractional Fourier transform spectrum: Figure 4 (a) is the spectrum of the simulated signal f(t), Figure 4 (b) is the angle α of the simulated signal f(t) opt = -0.01 fractional Fourier transform spectrum.

[0038] Figure 5 The following is a comparison diagram of the signal decomposed by the method of the present invention and the component signals f1(t), f2(t) and f3(t) of the simulation signal: Figure 5 (a) is the signal decomposed by the method of the present invention And the comparison diagram of the component signal f1(t), Figure 5(b) is the signal decomposed by the method of the present invention and the comparison diagram of the component signal f2(t), Figure 5 (c) is the signal decomposed by the method of the present invention Comparison diagram of component signal f3(t).

[0039] Figure 6 This is a comparison diagram of the signal reconstructed by the method of the present invention and the original simulation signal f(t). DETAILED DESCRIPTION

[0040] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0041] See Figures 1-6 , in order to facilitate analysis, we first introduce the definition of fractional Fourier transform. Any energy finite signal f(t)∈L 2 The fractional Fourier transform of (R) is defined as

[0042]

[0043] Where, Represents the fractional Fourier transform operator, kernel function The expression is

[0044]

[0045] Where k∈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. Correspondingly, the formula for the inverse transform of the fractional Fourier transform is

[0046]

[0047] In the formula, the superscript symbol * In particular, when α = π / 2, the fractional Fourier transform degenerates into the traditional Fourier transform.

[0048] In addition, to simplify the analysis, we also need to introduce the concept of convolution under fractional Fourier transform. The fractional convolution of any two energy-limited signals f(t) and g(t) is defined as

[0049]

[0050] Under the fractional Fourier transform, the fractional convolution satisfies

[0051]

[0052] Where G(ucscα) represents the Fourier transform of the signal g(t) (the transform element is scaled by cscα).

[0053] Furthermore, the definition of fractional wavelet transform is introduced. For any energy-limited signal f(t)∈L 2 The fractional wavelet transform of (R) is defined as

[0054]

[0055] Where, the kernel function ψ α,a,t The expression of (τ) is

[0056]

[0057] Where, the scale parameter a and the time shift parameter t satisfy: a∈R + , t∈R. According to the definition of fractional convolution, the definition of fractional wavelet transform can be rewritten as

[0058]

[0059] Therefore, combined with formula (5), it can also be rewritten as

[0060]

[0061] Where Ψ(ucscα) represents the Fourier transform of the mother wavelet function ψ(t) (the transform element is scaled by cscα). This shows that the fractional wavelet transform is essentially a set of multiplicative multiscale filters in the fractional Fourier transform domain. According to fractional Fourier transform theory, multiplicative filters in the fractional Fourier transform domain are linear time-varying filters, suitable for nonstationary signal processing.

[0062] Based on the above analysis, the present invention will now describe a data-driven fractional wavelet transform adaptive signal decomposition and reconstruction method.

[0063] Assume that the signal f(t)∈L 2 The fractional Fourier transform spectrum of (R) is divided into M continuous intervals, each of which corresponds to a modal component with a compactly supported fractional Fourier transform spectrum centered at a certain fractional frequency. represents the endpoint set of the interval, where n m ,n M ∈Z and n m <n M For the convenience of analysis, we agree that u0=0. So, if n<0, then u n<0; if n>0, then u n > 0. In most practical applications, the endpoint set V does not include u0 = 0. This is because real physical signals often contain more low-frequency components, and the endpoints of their fractional-order spectrum support intervals usually do not fall at the zero frequency point. is the set of interval endpoints that does not include u0 = 0. Based on this, the support interval of the fractional Fourier transform spectrum can be defined as

[0064]

[0065] Record Order represents the fractional Fourier transform spectrum support interval set, for the endpoint set The same definition can be used, but where n = n m ,···,n M And U -1 =[u -1 ,u1], at this time zero frequency u0∈U -1 The length of the support interval is expressed as |U n |=u n+1 -u n Therefore, the support interval of the fractional Fourier transform spectrum can be roughly summarized into the following four cases:

[0066] (1) Infinite support interval at both ends, that is, n m =-∞ and n M =+∞.

[0067] (2) Finite support interval, i.e. n m 、n M are all finite integers.

[0068] (3) Infinite support interval on the left, i.e. n m =-∞, n M is a finite integer; the leftmost support interval is expressed as

[0069]

[0070] (4) The infinite support interval on the right side, that is, n m is a finite integer, n M =+∞; the rightmost support interval is expressed as

[0071]

[0072] Although the support interval of the fractional Fourier transform spectrum can be summarized into the above four types in theory, the actual physical signals are often broadband and limited, and the case of infinite support interval usually does not occur, especially (1) infinite support interval at both ends. In view of this, the present invention only considers cases (2), (3), and (4). For the considered fractional Fourier transform spectrum support interval U n , if it is compactly supported, then its center is defined as

[0073]

[0074] In particular, if the fractional Fourier transform spectrum support interval considered is the infinite support interval on the left end Or in the infinite support interval on the right Then their centers are defined as

[0075]

[0076]

[0077] For endpoint sets The center of the support interval of the fractional Fourier transform spectrum under the same definition can also be used, except that c -1 , which is defined as c -1 =(u -1 +u1) / 2.

[0078] Based on the above description of the fractional Fourier transform spectrum support interval, the construction of the data-driven fractional wavelet is given below. For a given fractional Fourier transform spectrum support interval, each subinterval is represented as U n =[u n ,u n+1 ]. In order to fully reveal the time-frequency characteristics of the signal, the Gabor function is selected as the fractional mother wavelet because the Gabor function has the best time-frequency energy aggregation and can provide the best time-frequency distribution rate. Therefore, the expression of the fractional mother wavelet function can be obtained as

[0079]

[0080] According to the definition of fractional wavelet transform, the fractional Fourier transform domain form of the fractional mother wavelet function can be obtained by scaling the Fourier transform, that is,

[0081]

[0082] Where Ψ(u cscα) is the Fourier transform of the fractional mother wavelet function ψ(t) (the transform element is scaled by cscα). Further, the parameter ν = 0 is selected. Then there is

[0083]

[0084] Under this parameter, 99.999% of the energy of the fractional mother wavelet function is concentrated in the fractional Fourier transform domain interval [-sinα / 2, sinα / 2], and Ψ(0) = 1. Based on this, in the fractional Fourier transform domain finite support interval U n =[u n ,u n+1 ], the fractional wavelet basis function ψ used for signal decomposition n (t) is defined as

[0085]

[0086] Where, n (ucscα) is ψ n Fourier transform of (t) (the transformation element is scaled by cscα).

[0087] Similarly, on the left side, there is an infinite support interval The fractional wavelet basis function used for signal decomposition is defined as

[0088]

[0089] Similarly, there is an infinite support interval on the right side The fractional wavelet basis function used for signal decomposition is defined as

[0090]

[0091] Based on the above results, it can be calculated that the fractional wavelet basis function used for signal decomposition is The fractional Fourier transform domain form of satisfy

[0092]

[0093] This shows that the fractional wavelet basis function is not a set of orthogonal bases. Therefore, it is necessary to further construct the fractional wavelet basis function for signal reconstruction, which is recorded as Based on the above analysis, the expression of data-driven fractional wavelet transform is:

[0094]

[0095] Correspondingly, the inverse transform formula of the data-driven fractional wavelet transform is

[0096]

[0097] The reconstructed fractional wavelet basis function is given below The conditions are satisfied. Perform the fractional Fourier transform on both sides of Equation (21) with respect to time t, and we get

[0098]

[0099] Similarly, if we perform fractional Fourier transform on both sides of equation (22) with respect to time t, we have

[0100]

[0101] Where, Φ n (ucscα) is the fractional mother wavelet function φ n The Fourier transform of (t) is performed (the transform element is scaled by cscα). Therefore, the fractional wavelet basis function used for signal reconstruction is and fractional wavelet basis functions for signal decomposition satisfy

[0102]

[0103] Based on this and combined with formula (20), we can know that for a given fractional wavelet basis function used for decomposition The fractional wavelet basis function used for reconstruction can be defined That is, its satisfaction

[0104]

[0105] The present invention will be described in detail below with reference to specific examples.

[0106] The present invention proposes a data-driven fractional wavelet transform adaptive signal decomposition and reconstruction method, which includes the following steps:

[0107] Step 1: Given the signal to be decomposed f(t), calculate the angle α corresponding to the fractional Fourier transform domain where the energy of the signal f(t) is optimally concentrated. opt ,Right now

[0108]

[0109] Step 2: Calculate the α where the signal to be decomposed f(t) is best concentrated in its energy opt The support interval U in the angle fractional Fourier transform domain n =[u n ,u n+1 ], where n m ≤n≤n M and n,n m ,n M ∈Z; at the same time, calculate the support interval center c n ;

[0110] Step 3: According to the specific type of the calculated support interval, calculate the center of the support interval and construct the fractional wavelet basis function for signal decomposition. The following three cases are explained:

[0111] (1) If the signal to be decomposed f(t) is at the α where its energy is best concentrated opt The support interval in the angle fractional Fourier transform domain belongs to the finite support interval, that is, Where -∞<n m ≤n≤n M <+∞ and n,n m ,n M ∈Z; then

[0112]

[0113] (2) If the signal to be decomposed f(t) is at the α where its energy is best concentrated opt The support interval on the angle fractional Fourier transform domain belongs to the infinite support interval on the left end, that is, where -∞=n m ≤n≤n M <+∞ and n,n m ,n M ∈Z, then the finite support interval The center of and its corresponding fractional wavelet basis function expressions are:

[0114]

[0115] However, for the infinite support interval on the left Then we have:

[0116]

[0117]

[0118] (3) If the signal to be decomposed f(t) is at the α where its energy is best concentrated opt The support interval on the angle fractional Fourier transform domain belongs to the infinite support interval on the right end Where -∞<n m ≤n≤n M =+∞ and n,n m ,n M ∈Z, then the finite support interval The center of and its corresponding fractional wavelet basis function expressions are:

[0119]

[0120] However, for the rightmost infinite support interval Then we have:

[0121]

[0122]

[0123] Step 4: Use the inverse transform of Fourier transform (the transform element is scaled cscα opt ), and obtain the time domain form of the fractional-order wavelet basis function for signal decomposition based on data driving, that is,

[0124] n m ≤n≤n M and n,n m ,n M ∈Z

[0125] Step 5: Solve the fractional-order wavelet basis function ψ for signal reconstruction based on the fractional-order wavelet basis function constructed for signal decomposition in the fractional-order Fourier transform domain. n (t), i.e.

[0126] n m ≤n≤n M and n,n m ,n M ∈Z

[0127] Step 6: Use the inverse transform of Fourier transform (the transform element is scaled cscα opt The time domain form of the fractional-order wavelet basis function for signal reconstruction is obtained by stretching, namely:

[0128] n m ≤n≤n M and n,n m ,n M ∈Z

[0129] Step 7: Use the constructed fractional wavelet basis function ψ n (t), the signal f(t) is decomposed by fractional convolution operation, and the decomposed signal is:

[0130] n m ≤n≤n M and n,n m ,n M ∈Z

[0131] Step 8: Based on the decomposed signal Using the constructed fractional wavelet basis function φ n (t), the signal is completely reconstructed through fractional-order convolution operation, that is:

[0132]

[0133] The effect of the present invention can be further illustrated by the following simulation:

[0134] Simulation signal f(t)=f1(t)+f2(t)+f3(t), where each component signal is: f1(t)=cos(30πt+50t 2 ), f2(t)=cos(70πt+50t 2 ), f3(t)=cos(100πt+50t 2 ). The time domain waveform of the simulation signal f(t) is as follows Figure 2 As shown. The time domain waveform of the signal component f1(t) is shown as Figure 3 As shown in (a), the time domain waveform of the signal component f2(t) is as follows Figure 3 As shown in (b), the time domain waveform of the signal component f3(t) is as follows Figure 3 (c) shown.

[0135] The spectrum of the simulated signal f(t) is as follows Figure 4 As shown in (a), it can be found that the energy of the simulated signal f(t) is non-aggregated in the frequency domain, and the frequency domain support intervals corresponding to the three component signals cannot be effectively extracted. Figure 4 (b) gives the simulated signal f(t) with an angle of α opt =-0.01 fractional Fourier transform spectrum. It can be seen that the support intervals corresponding to the three component signals can be effectively extracted in the fractional Fourier transform domain. The method of the present invention can be used to obtain and The comparison results of the three decomposed signals with the three components of the simulation signal f1(t), f2(t) and f3(t) are as follows Figure 5 shown. Figure 5 It shows that the method of the present invention can effectively realize signal decomposition. Finally, the decomposition results are used and Perform signal reconstruction, and the reconstructed signal is as follows Figure 6 It can be seen that the method of the present invention can realize the reconstruction of the signal.

Claims

1. A data-driven fractional wavelet transform adaptive signal decomposition and reconstruction method, characterized by: The method comprises the following steps: Step 1: Given the signal to be decomposed f(t), calculate the angle α corresponding to the fractional Fourier transform domain where the energy of the signal f(t) is optimally concentrated. opt ,Right now Step 2: Calculate the α where the signal to be decomposed f(t) is best concentrated in its energy opt The support interval U in the angle fractional Fourier transform domain n =[u n ,u n+1 ], where n m ≤n≤n M and n,n m ,n M ∈Z; at the same time, calculate the support interval center c n ; Step 3: According to the specific type of the calculated support interval, calculate the center of the support interval and construct the fractional wavelet basis function for signal decomposition. The following three cases are explained: (1) If the signal to be decomposed f(t) is at the α where its energy is best concentrated opt The support interval in the angle fractional Fourier transform domain belongs to the finite support interval, that is, Where -∞<n m ≤n≤n M <+∞ and n,n m ,n M ∈Z; then (2) If the signal to be decomposed f(t) is at the α where its energy is best concentrated opt The support interval on the angle fractional Fourier transform domain belongs to the infinite support interval on the left end, that is, where -∞=n m ≤n≤n M <+∞ and n,n m ,n M ∈Z, then the finite support interval The center of and its corresponding fractional wavelet basis function expressions are: However, for the infinite support interval on the left Then we have: (3) If the signal to be decomposed f(t) is at the α where its energy is best concentrated opt The support interval on the angle fractional Fourier transform domain belongs to the infinite support interval on the right end Where -∞<n m ≤n≤n M =+∞ and n,n m ,n M ∈Z, then the finite support interval The center of and its corresponding fractional wavelet basis function expressions are: However, for the rightmost infinite support interval Then we have: Step 4: Use the inverse transform of Fourier transform to obtain the time domain form of the fractional wavelet basis function for signal decomposition based on data drive, that is, n m ≤n≤n M and n,n m ,n M ∈Z Step 5: Solve the fractional-order wavelet basis function ψ for signal reconstruction based on the fractional-order wavelet basis function constructed for signal decomposition in the fractional-order Fourier transform domain. n (t), i.e. n m ≤n≤n M and n,n m ,n M ∈Z Step 6: Use the inverse transform of Fourier transform to obtain the time domain form of the fractional-order wavelet basis function used for signal reconstruction, that is: n m ≤n≤n M and n,n m ,n M ∈Z Step 7: Use the constructed fractional wavelet basis function ψ n (t), the signal f(t) is decomposed by fractional convolution operation, and the decomposed signal is: n m ≤n≤n M and n,n m ,n M ∈Z Step 8: Based on the decomposed signal Using the constructed fractional wavelet basis function φ n (t), the signal is completely reconstructed through fractional-order convolution operation, that is:

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