Empirical fractional Fourier transform domain signal decomposition method

Abstract

The invention provides an empirical fractional Fourier transform domain signal decomposition method, which comprises the following steps of: 1, segmenting a fractional Fourier spectrum of an original signal f (t) to obtain N fractional Fourier spectrum segments; step 2, constructing a zero-phase filter # imgabs0 # for signal decomposition based on the fractional order Fourier spectrum segmentation segments; step 3, filtering the original signal f (t) based on the zero-phase filter # imgabs1 # to obtain a signal decomposition result of each fractional order Fourier spectrum segmentation segment; and 4, calculating the sum of all decomposition results to obtain a reconstructed signal. According to the empirical fractional order Fourier transform domain signal decomposition method provided by the invention, the problem that components of the signal are mutually overlapped and cannot be separated due to Fourier transform domain signal energy diffusion can be avoided.

Description

Technical Field

[0001] The invention relates to an empirical fractional Fourier transform domain signal decomposition method, belonging to the technical field of signal and information processing. Background Art

[0002] Classical signal analysis is mostly based on basis function expansion, and requires the signal to have a stable characteristic. It lacks the ability to adapt to the complex changes in signal form and is difficult to accurately extract the characteristic information contained in the signal. How to effectively extract characteristic information from complex non-stationary signals has become a fundamental problem in signal processing.

[0003] The time-varying Fourier transform is a typical feature of non-stationary signals. In order to effectively analyze complex non-stationary signals, people have proposed a series of signal analysis methods that combine time and frequency, such as short-time Fourier transform, wavelet transform, Wigner-Ville distribution, etc. However, these methods are based on integral transforms, and require artificial selection or design of appropriate basis functions according to signal characteristics to match signal characteristic components, and cannot adapt to changes in signal morphology. In order to overcome this problem, adaptive signal decomposition methods represented by empirical mode decomposition came into being.

[0004] Empirical mode decomposition does not need to design basis functions that meet the signal characteristics based on prior knowledge. It belongs to a posteriori decomposition. The decomposition process is completely driven by data and has strong signal adaptability. However, with the continuous deepening of research and application, empirical mode decomposition has also exposed many shortcomings, such as the lack of rigorous mathematical theoretical support, the existence of modal aliasing and endpoint effects. In view of this, under the framework of wavelet theory, people proposed the empirical wavelet transform. This transform has the advantages of both empirical mode decomposition and wavelet analysis. Based on the characteristics of the data itself, it adaptively divides the Fourier spectrum of the original signal, constructs a framework wavelet filter on each segmentation interval, and uses it to filter the original signal to obtain the modal component of the amplitude-frequency modulation signal with a compactly supported Fourier spectrum, thereby realizing the 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. In addition, based on Fourier transform theory, people proposed the Fourier decomposition method. This method decomposes the signal into a small number of Fourier intrinsic frequency band functions through the generalized Fourier expansion of a time series with variable amplitude and variable frequency, thereby obtaining a stable signal decomposition. Compared with the empirical mode decomposition, the Fourier decomposition method is supported by the zero-phase filter and Fourier transform theory. However, due to the defects of the Fourier spectrum segmentation technology, meaningless trivial components may appear in the empirical wavelet transform. At the same time, due to the existence of the wavelet filter transition band, the transformation result may also have the problem of modal aliasing. Similarly, the Fourier decomposition method may obtain inconsistent decomposition results based on two frequency scanning techniques, namely the low-to-high technique and the high-to-low technique, and it is impossible to determine which decomposition result is correct. In addition, both frequency scanning techniques are iterative and have high complexity. Therefore, by combining the ideas of the empirical wavelet transform and the Fourier decomposition method, the empirical Fourier decomposition method is proposed. This method not only overcomes the problems of trivial components and modal aliasing in the empirical wavelet transform, but also solves the problem of inconsistent decomposition results obtained by using different frequency scanning techniques in Fourier decomposition.

[0005] However, the zero-phase filter is essentially a linear time-invariant filter in the Fourier transform domain, which is only suitable for stationary signals with optimal energy concentration in the Fourier transform domain. For signals with non-optimal energy concentration in the Fourier transform domain, the result of empirical Fourier decomposition is not optimal. Therefore, the present invention proposes an empirical fractional Fourier transform domain signal decomposition method. Summary of the invention

[0006] The present invention aims to solve the problem that in the prior art, when analyzing complex non-stationary signals, it is necessary to manually select or design appropriate basis functions according to signal characteristics to match signal characteristic components, and the problem that the change of signal morphology cannot be adaptively adapted is proposed.

[0007] The technical solution adopted by the present invention to solve the above problems is: the present invention proposes an empirical fractional Fourier transform domain signal decomposition method, comprising:

[0008] Step 1: Segment the fractional Fourier spectrum of the original signal f(t) to obtain N fractional Fourier spectrum segmentation segments;

[0009] Step 2: Construct a zero-phase filter for signal decomposition based on the fractional Fourier spectrum segmentation

[0010] Step 3: Based on zero-phase filter Perform filtering on the original signal f(t) to obtain the decomposition result of each fractional Fourier spectrum segment signal;

[0011] Step 4: Calculate the sum of all decomposition results to obtain the reconstructed signal.

[0012] Optionally, the step of obtaining N fractional-order Fourier spectrum segmentation segments in step 1 includes:

[0013] Step 1.1: Define the components to be decomposed in the original signal f(t) as N, and calculate the angle α corresponding to the fractional Fourier transform domain where the energy of the original signal f(t) is optimally concentrated opt ;

[0014] Step 1.2: Calculate the original signal f(t) at the optimal energy focusing angle α opt The fractional Fourier transform spectrum of

[0015] Step 1.3: Use segmentation techniques to segment the original signal f(t) at the angle α where the energy is best concentrated opt The fractional Fourier spectrum of Calculate the support interval U of the N segments of the fractional Fourier spectrum of the original signal f(t) n =[u n ,u n+1 ], where 0≤n≤N-1 and

[0016] The angle α corresponding to the fractional Fourier transform domain where the energy of the original signal f(t) is optimally concentrated opt The expression is:

[0017]

[0018] Optionally, the step of calculating the support intervals of the N segmented segments of the fractional-order Fourier spectrum of the original signal f(t) in step 1.3 includes:

[0019] Step 1.3.1: The original signal f(t) is concentrated at the angle α where the energy is best concentratedopt The fractional Fourier spectrum of During the segmentation process, identify the fractional Fourier spectrum of the original signal The maximum point and endpoint u L 、u N The amplitude at ;

[0020] Step 1.3.2: Convert the fractional Fourier spectrum The values ​​are sorted in descending order and stored as a sequence. The fractional frequencies corresponding to the first N maxima in the sequence are represented as a vector [Ω1,Ω2,...Ω N ];

[0021] Step 1.3.3: Define Ω0 = u L and Ω N+1 =u H , we get the vector [Ω0,Ω2,...Ω N+1 ], using the vector [Ω0,Ω2,...Ω N+1 ] Determine the support interval of N fractional-order spectrum segmentation segments;

[0022] The expression of the boundary of the support interval of N fractional-order spectrum segmentation segments is:

[0023]

[0024] In formula (2), is the signal f(t) in Ω n and Ω n+1 The magnitude of the fractional Fourier spectrum between .

[0025] Optional, zero-phase filter for signal decomposition in step 2 The expression of fractional Fourier transform domain is:

[0026]

[0027] Optionally, the step of obtaining the decomposition result of each fractional-order Fourier spectrum segment signal in step 3 includes:

[0028] Step 3.1: Based on zero-phase filter Perform fractional convolution operation to obtain the angle α where the energy of the original signal f(t) is optimally concentrated opt The decomposition result of each segment in the fractional Fourier transform domain in the fractional Fourier spectrum below;

[0029] Step 3.2: Obtain the time domain expression of the decomposition result through inverse fractional Fourier transform;

[0030] The expression of the decomposition result of each segment in the fractional Fourier transform domain is:

[0031]

[0032] The expression of the decomposition result in time domain is:

[0033]

[0034] In formula (5), 0≤n≤N-1 and

[0035] Optionally, the expression of the reconstructed signal in step 4 is:

[0036]

[0037] In formula (6), 0≤n≤N-1 and

[0038] The beneficial effects of the present invention are:

[0039] 1. Compared with the existing empirical Fourier transform domain signal decomposition method, the empirical fractional-order Fourier transform domain signal decomposition method proposed in the present invention can avoid the problem that the signal energy diffusion in the Fourier transform domain causes the signal components to overlap with each other and cannot be separated.

[0040] 2. The error between the component signal extracted by the empirical fractional Fourier transform domain signal decomposition method of the present invention and the component signal of the simulated signal is less than 1.47×10 -5 , which is much smaller than the existing empirical Fourier decomposition method, and improves the decomposition accuracy of the signal without increasing the computational complexity. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 A flow chart of an empirical fractional Fourier transform domain signal decomposition method provided by the present invention;

[0042] Figure 2 A time domain waveform diagram of the simulation signal f(t) provided by the present invention;

[0043] Figure 3 The time domain waveform diagram of each component signal f1(t), f2(t), f3(t) and f4(t) of the simulation signal f(t) provided by the present invention;

[0044] Figure 3 middle, 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), Figure 3 (d) is the time domain waveform of the component signal f4(t);

[0045] Figure 4 The Fourier spectrum and angle α of the simulation signal f(t) provided by the present invention opt = -0.0997 fractional Fourier spectrum;

[0046] Figure 4 middle, Figure 4 (a) is the Fourier spectrum of the simulated signal f(t), Figure 4 (b) is the simulated signal f(t) at angle α opt = -0.0997 fractional Fourier spectrum;

[0047] Figure 5 A comparison diagram of the time domain waveforms of the decomposed component signals provided by the present invention and the component signals f1(t), f2(t), f3(t) and f4(t) of the simulation signal f(t):

[0048] Figure 5 middle, Figure 5 (a) is the component signal decomposed by the method of the present invention And the comparison diagram of the time domain waveform of the component signal f1(t), Figure 5 (b) is the component signal decomposed by the method of the present invention And the comparison diagram of the time domain waveform of the component signal f2(t), Figure 5 (c) is the component signal decomposed by the method of the present invention And the comparison diagram of the time domain waveform of the component signal f3(t), Figure 5 (d) is the component signal decomposed by the method of the present invention and the comparison diagram of the time domain waveform of the component signal f4(t);

[0049] Figure 6 The reconstructed signal provided by the present invention Comparison with the time domain waveform of the original simulation signal f(t). DETAILED DESCRIPTION

[0050] Combination Figure 1 This embodiment is described. First, this embodiment introduces the definition of fractional Fourier transform for analyzing non-stationary signals, and includes the following steps:

[0051] For any energy-finite signal f(t)∈L 2 (R), the fractional Fourier transform is defined as:

[0052]

[0053] In formula (1), Fα is the fractional Fourier transform operator, F α (u) is the fractional Fourier spectrum of the signal f(t), K α(u, t) is the kernel function, kernel function K α The expression of (u,t) is:

[0054]

[0055] In formula (2), k∈Z, α is the angle of fractional Fourier transform, variable u is usually called fractional frequency, and the coordinate axis where it is located is usually called fractional Fourier transform domain. Correspondingly, the formula of inverse fractional Fourier transform is:

[0056]

[0057] In formula (3), the superscript symbol * indicates the conjugate operation;

[0058] When α=π / 2, the fractional Fourier transform degenerates into the Fourier transform.

[0059] In order to facilitate subsequent analysis, this implementation also introduces the concept of fractional-order filtering. Classical convolution is the theoretical basis of Fourier transform domain filtering. Correspondingly, fractional-order filtering is based on fractional-order convolution. The fractional-order convolution of any two energy-limited signals f(t) and h(t) is defined as:

[0060]

[0061] Under the fractional Fourier transform, the fractional convolution satisfies

[0062]

[0063] In formula (5), H(u cscα) is the Fourier transform of h(t), and its transform element is scaled by cscα.

[0064] According to the time domain and fractional Fourier transform domain expressions of fractional convolution, after the excitation f(t) passes through the linear system h(t), the response y(t) can be expressed in the fractional Fourier transform domain as follows:

[0065]

[0066] In formula (6), Y α (u) is the fractional Fourier transform of y(t) at the fractional Fourier transform angle α.

[0067] like Figure 1 As shown, based on the above analysis, the steps of an empirical fractional Fourier transform domain signal decomposition method described in this embodiment include:

[0068] S1: Calculate the fractional Fourier transform F of the signal to be analyzed f(t) based on formula (1) α (u);

[0069] S2: Calculate the angle α corresponding to the fractional Fourier transform domain where the signal f(t) energy is optimally concentrated opt ;

[0070] The angle α corresponding to the fractional Fourier transform domain where the signal f(t) energy is optimally concentrated opt The expression is:

[0071]

[0072] S3: Calculate the fractional Fourier spectrum of the signal f(t) at the optimal energy concentration angle

[0073] S4: Segmentation of the Fractional Fourier Spectrum Calculate the boundary u of the segment n ;

[0074] S401: The original signal f(t) is at the angle α where the energy is best concentrated opt The fractional Fourier spectrum of During the segmentation process, identify the fractional Fourier spectrum of the original signal The maximum point and endpoint u L 、u N The amplitude at ;

[0075] S402: fractional Fourier spectrum The values ​​are sorted in descending order and stored as a sequence. The fractional frequencies corresponding to the first N maxima in the sequence are represented as a vector [Ω1,Ω2,...Ω N ];

[0076] S403: define Ω0=u L and Ω N+1 =u H , we get the vector [Ω0,Ω2,...Ω N+1 ], using the vector [Ω0,Ω2,...Ω N+1 ] Determine the support interval U of N fractional-order spectrum segment segments n =[u n ,u n+1 ], where 0≤n≤N-1 and

[0077] The expression of the boundary of the support interval of N fractional-order spectrum segmentation segments is:

[0078]

[0079] In formula (8), is the signal f(t) in Ω n and Ωn+1 The magnitude of the fractional Fourier spectrum between .

[0080] S5: According to the boundary u of the segment n Constructing a Zero-Phase Filter

[0081] Zero Phase Filter The expression is:

[0082]

[0083] S6: Use zero-phase filter Filter the signal f(t) in the fractional Fourier transform domain to obtain the fractional Fourier spectrum of the decomposed signal;

[0084] The expression of the filtering process of each zero-phase filter on the original signal f(t) in the fractional Fourier transform domain is:

[0085]

[0086] S7: Obtain the time domain signal of the decomposition result through inverse Fourier transform;

[0087] The expression of the time domain signal of the decomposition result is:

[0088]

[0089] In formula (11), 0≤n≤N-1 and

[0090] S8: superimpose the signals of the decomposition results to obtain the reconstructed signal;

[0091] The expression of the reconstructed signal is:

[0092]

[0093] In formula (12), 0≤n≤N-1 and

[0094] When α=π / 2, the empirical fractional Fourier transform domain signal decomposition degenerates into the empirical Fourier transform domain signal decomposition.

[0095] Example

[0096] Combination Figure 2-6 This embodiment is described.

[0097] In this embodiment, the expression of the simulation signal is f(t)=f1(t)+f2(t)+f3(t)+f4(t), where each component signal is:

[0098]

[0099] like Figure 2 As shown, Figure 2 The time domain waveform of the simulated signal f(t) is given, and the time domain waveform of the component signal f1(t) is shown as Figure 3 As shown in (a), the time domain waveform of the component signal f2(t) is Figure 3 As shown in (b), the time domain waveform of the component signal f3(t) is Figure 3 As shown in (c), the time domain waveform of the component signal f4(t) is Figure 3 (d) as shown.

[0100] The Fourier 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 simulation signal f(t) is non-aggregated in the Fourier transform domain, and the support intervals corresponding to the four component signals cannot be extracted. Figure 4 (b) shows the simulated signal f(t) at an angle α opt =-0.0997 fractional Fourier spectrum, it can be seen that the support intervals corresponding to the four component signals can be effectively extracted in the fractional Fourier transform domain.

[0101] This embodiment can extract four components from the signal f(t): and The time domain comparison of the component signal of the simulated signal f(t) is Figure 5 As shown in the figure, the mean square errors of the component signals extracted by the empirical Fourier transform domain signal decomposition method and the simulated signal f(t) are 0.18, 1.19, 0.81 and 0.82 respectively; the mean square errors of the component signals extracted by the empirical fractional Fourier transform domain signal decomposition method and the simulated signal f(t) are 1.47×10 -5 ,8.47×10 -6 , 1.53×10 -6 and 1.35×10 -6 , it can be concluded that compared with the Fourier transform domain, the signal energy is more concentrated in the fractional Fourier transform domain; compared with the empirical Fourier transform domain signal decomposition method, the empirical fractional Fourier transform domain signal decomposition method has a better segmentation effect, and the decomposed component signal is closer to the original component signal. The reconstructed signal is as follows Figure 6 shown.

[0102] The above is only a preferred embodiment of the present invention and does not limit the present invention in any form. Although the present invention has been disclosed as a preferred embodiment as above, it is not used to limit the present invention. Any technician familiar with this profession can make some changes or modify the technical contents disclosed above into equivalent embodiments without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent replacement and improvement made to the above embodiments without departing from the content of the technical solution of the present invention, based on the technical essence of the present invention, within the spirit and principles of the present invention, still fall within the protection scope of the technical solution of the present invention.

Claims

1. An empirical fractional Fourier transform domain signal decomposition method, characterized in that: The steps of the empirical fractional Fourier transform domain signal decomposition method include: Step 1: Segment the fractional Fourier spectrum of the original signal f(t) to obtain N fractional Fourier spectrum segmentation segments; Step 2: Construct a zero-phase filter for signal decomposition based on the fractional Fourier spectrum segmentation Step 3: Based on the zero-phase filter Perform filtering on the original signal f(t) to obtain the decomposition result of each fractional Fourier spectrum segment signal; Step 4: Calculate the sum of all decomposition results to obtain the reconstructed signal.

2. The empirical fractional Fourier transform domain signal decomposition method according to claim 1, characterized in that: The steps of obtaining N fractional-order Fourier spectrum segmentation segments in step 1 include: Step 1.1: Define the components to be decomposed in the original signal f(t) as N, and calculate the angle α corresponding to the fractional Fourier transform domain where the energy of the original signal f(t) is optimally concentrated opt ; Step 1.2: Calculate the optimal energy concentration angle α of the original signal f(t) opt The fractional Fourier transform spectrum of Step 1.3: Use segmentation techniques to segment the original signal f(t) at the angle α where the energy is best concentrated opt The fractional Fourier spectrum of Calculate the support interval U of the N segments of the fractional Fourier spectrum of the original signal f(t) n =[u n ,u n+1 ], where 0≤n≤N-1 and The angle α corresponding to the fractional Fourier transform domain where the energy of the original signal f(t) is optimally concentrated opt The expression is:

3. The empirical fractional Fourier transform domain signal decomposition method according to claim 2, characterized in that: The calculation steps of the support intervals of the N segments of the fractional Fourier spectrum of the original signal f(t) in step 1.3 include: Step 1.3.1: The original signal f(t) is concentrated at the angle α where the energy is best concentrated opt The fractional Fourier spectrum of During the segmentation process, identify the fractional Fourier spectrum of the original signal The maximum point and endpoint u of L 、u N The amplitude at ; Step 1.3.2: Convert the fractional Fourier spectrum The values ​​are sorted in descending order and stored as a sequence. The fractional frequencies corresponding to the first N maxima in the sequence are represented as a vector [Ω1,Ω2,...Ω N ]; Step 1.3.3: Define Ω0 = u L and Ω N+1 =u H , we get the vector [Ω0,Ω2,...Ω N+1 ], using the vector [Ω0,Ω2,...Ω N+1 ] Determine the support interval of N fractional-order spectrum segmentation segments; The expression of the boundary of the support interval of N fractional-order spectrum segmentation segments is: In formula (2), is the signal f(t) in Ω n and Ω n+1 The magnitude of the fractional Fourier spectrum between .

4. The empirical fractional Fourier transform domain signal decomposition method according to claim 1, characterized in that: Zero-phase filter used for signal decomposition in step 2 The expression of the fractional Fourier transform domain is:

5. The method for decomposing a signal in the empirical fractional Fourier transform domain according to claim 1, characterized in that: The steps of obtaining the decomposition result of each fractional-order Fourier spectrum segment signal in step 3 include: Step 3.1: Based on the zero-phase filter Perform fractional convolution operation to obtain the angle α where the energy of the original signal f(t) is optimally concentrated opt The decomposition result of each segment in the fractional Fourier transform domain in the fractional Fourier spectrum below; Step 3.2: Obtain the time domain expression of the decomposition result through inverse fractional Fourier transform; The expression of the decomposition result of each segment in the fractional Fourier transform domain is: The expression of the decomposition result in time domain is: In formula (5), 0≤n≤N-1 and 6. The method for decomposing a signal in the empirical fractional Fourier transform domain according to claim 1, characterized in that: The expression of the reconstructed signal in step 4 is: In formula (6), 0≤n≤N-1 and

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