Variational mode decomposition method and device based on orthogonal criterion, and computing equipment
By using a variational mode decomposition method based on orthogonality criteria, the mode center frequency and bandwidth are adaptively adjusted, which solves the problems of insufficient parameter dependence and anti-aliasing performance of VMD, and realizes reliable and accurate signal decomposition in complex noise environments.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- INST OF ACOUSTICS CHINESE ACAD OF SCI
- Filing Date
- 2025-12-26
- Publication Date
- 2026-05-08
AI Technical Summary
Existing variational mode decomposition (VMD) methods are highly dependent on parameter selection, resulting in unstable signal decomposition performance and insufficient resistance to mode mixing in complex noise environments.
A variational mode decomposition method based on orthogonality criterion is adopted. By performing background equalization on the input signal, extracting empirical spectral trends, constructing a variational model based on orthogonality, adaptively adjusting the center frequency and bandwidth of the modes, and performing iterative optimization using the Lagrange multiplier method and the alternating direction multiplier method, signal decomposition is achieved.
It achieves adaptability and accuracy in signal decomposition under complex noise environments, improves anti-modal aliasing capability, and dynamically balances decomposition accuracy and flexibility.
Smart Images
Figure CN121996910A_ABST
Abstract
Description
Technical Field
[0001] This application relates to the field of signal processing technology, and in particular to a variational mode decomposition method, apparatus and computing device based on orthogonality criteria. Background Technology
[0002] With the increasing demand for weak signal detection and non-stationary signal analysis in complex environments, adaptive mode decomposition (AMD) methods, represented by empirical mode decomposition (EMD), empirical wavelet transform (EVT), and variational mode decomposition (VMD), have become important research directions. EMD achieves adaptive signal decomposition in the time domain by successively extracting the intrinsic mode functions of the signal, exhibiting good time-frequency adaptability. However, it lacks a mathematical foundation and suffers from problems such as mode aliasing, endpoint effects, sensitivity to sampling and noise, and unstable decomposition results. EWT achieves signal decomposition by adaptively dividing the frequency band and constructing filter banks in the spectrum, offering better theoretical interpretability than EMD. However, spectrum segmentation relies on the detection of local extrema, is sensitive to noise, and the choice of filter boundaries significantly affects the decomposition accuracy.
[0003] Virtual Mode Decomposition (VMD) estimates and reconstructs the center frequency of a signal component by solving a variational constraint optimization problem based on the narrowband conditions of the signal components. This method is built upon a clear mathematical foundation, including Wiener filtering, Hilbert transform, and frequency mixing, forming a novel, adaptive, and non-recursive theoretical framework for signal decomposition. Compared to methods like EMD, VMD is more efficient and exhibits better resistance to mode aliasing and noise robustness, thus attracting widespread attention from academic and industrial communities both domestically and internationally. It has been successfully applied in various fields, including mechanical fault diagnosis, biomedical signal analysis, image signal analysis, and radar / sonar signal processing.
[0004] However, the performance of VMD is closely related to the selection of model parameters, including the number of decomposed modes, bandwidth constraint parameters, and initial center frequencies of the modes. Therefore, determining the optimal variational model parameters is crucial for the signal decomposition effect of VMD and its application in various fields. Summary of the Invention
[0005] Firstly, this application provides a variational mode decomposition method based on orthogonality criteria; applied to multi-component non-stationary signal processing in complex noise environments, the method includes: performing background equalization on the input signal and extracting empirical spectrum trends; the empirical spectrum trends indicate the overall fluctuation of the target signal's spectral energy; the target signal includes mechanical vibration signals, biomedical signals, electromagnetic wave signals, acoustic wave signals, or artificial signals; determining initial parameters based on the empirical spectrum trends, the initial parameters including the number of decomposition modes, the initial center frequency of each mode, and the initial bandwidth; constructing an orthogonal constrained variational model based on the orthogonality of the input signal in the signal space, the orthogonality being reflected in the relationship between each mode and its corresponding residual; iteratively updating each mode, the center frequency of each mode, and the bandwidth based on the orthogonal constrained variational model to obtain the signal decomposition result.
[0006] In some possible implementations, the orthogonal constrained variational model includes:
[0007]
[0008] in, For the mode to be estimated, Let be the center frequency of the mode to be estimated; This is a modal bandwidth constraint parameter used to balance the bandwidth of the mode with the fidelity of signal reconstruction; Input signal In terms of modality The orthogonal projection onto a specific basis in the spanned signal space; For residual terms; For the first The residuals corresponding to each mode; For the first A filter that corresponds to the residual of each mode.
[0009] In some possible implementations, the method further includes: solving the optimization problem of the orthogonal constrained variational model using the Lagrange multiplier method to obtain the augmented Lagrange equation of the orthogonal constrained variational model as follows:
[0010]
[0011] in, Represents the Lagrange multipliers; This represents inner product operations.
[0012] In some possible implementations, the orthogonal constrained variational model is used to iteratively update each mode, its center frequency, and its bandwidth, including: decomposing the optimization problem of the orthogonal constrained variational model into a series of sub-optimization problems using the alternating direction multiplier method, and solving them iteratively in the frequency domain; in the... In the nth iteration, the th Each mode and its center frequency are updated in the frequency domain as follows:
[0013]
[0014]
[0015] in, , , Represent , as well as Fourier transform, For the first A filter that corresponds to the residual of each mode.
[0016] In some possible implementations, the modes, their center frequencies, and bandwidths are iteratively updated based on the orthogonal constrained variational model, including: according to the... The initial bandwidth of each sub-band is determined. The initial bandwidth constraint parameters for each mode are updated using a coarse-to-fine adaptive mechanism. In the nth iteration, the 1st The bandwidth constraint parameters corresponding to each mode are:
[0017]
[0018] in, , They are respectively , Fourier transform; This indicates taking the real part of a complex number.
[0019] In some possible implementations, the modes, their center frequencies, and bandwidths are iteratively updated based on the orthogonal constrained variational model, including: in the first... In the next iteration, the Lagrange multipliers are updated as follows:
[0020]
[0021] in, for Fourier transform; is the step size parameter of the Lagrange multiplier.
[0022] In some possible implementations, the first A filter that corresponds to the residual of each mode. The frequency response is:
[0023]
[0024] in, For a smooth transition function; the first Effective bandwidth of each mode for:
[0025]
[0026] In some possible implementations, the modes, their center frequencies, and bandwidths are iteratively updated based on the orthogonal constrained variational model, including:
[0027] The sum of the variances of all modes in two consecutive iterations is used as the algorithm's convergence evaluation metric. When the convergence evaluation metric is less than the preset convergence parameter... Under the following conditions, the algorithm convergence condition is met:
[0028]
[0029] If the algorithm convergence condition is met, the process is terminated. Iterative updates of the spectrum, center frequency, and bandwidth of each mode.
[0030] In some possible implementations, initial parameters are determined based on empirical spectral trends. These initial parameters include the number of decomposed modes, the initial center frequency of each mode, and the initial bandwidth. This includes: extracting the observed signal spectral trend based on the amplitude spectrum of the input signal, where the observed signal spectral trend includes a combination of the target signal spectral trend varying with frequency and the background noise spectral trend; estimating the mean background noise based on the amplitude spectrum of the input signal; extracting the background noise spectral trend based on the mean background noise; calculating the difference between the observed signal spectral trend and the background noise spectral trend to obtain the empirical spectral trend; the empirical spectral trend indicates the overall fluctuation of the target signal's spectral energy; and traversing the empirical spectral trend... The first critical point is determined. Sub-bands; the first critical point includes the boundary frequency of the spectrum, the local minimum point of the empirical spectrum trend, or the gradient abrupt change point; For natural numbers greater than or equal to 1; based on the frequency band division results, traverse the empirical spectrum trend. The second critical point is determined. The first center frequency; the second critical point includes the spectral peak frequency, the peak point of the empirical spectral trend, or the gradient abrupt change point; based on the number of sub-bands. The number of modes is determined by the center frequency corresponding to each sub-band. The initial center frequency of the mode; the initial bandwidth of each mode is determined based on the boundary frequencies of each sub-band.
[0031] The method provided in this application embodiment determines the input signal based on the empirical spectrum trend by performing background equalization and empirical spectrum trend extraction on the input signal. The boundary frequency and number of signal components of each sub-band. By using the center frequency of each component, adaptive frequency band division and center frequency estimation of signal components can be achieved.
[0032] The variational mode decomposition method based on orthogonality criteria provided in this application adaptively adjusts the bandwidth constraint parameters of each mode according to the measure of orthogonality in the frequency domain during the iterative optimization process. This mechanism overcomes the limitations of traditional VMD where bandwidth is globally fixed or manually set, achieving a dynamic balance between decomposition accuracy and flexibility.
[0033] The variational mode decomposition method based on orthogonality criterion provided in this application adaptively determines the number of decomposed modes and the initial center frequency of the modes. Based on the orthogonality criterion of modes, it realizes dynamic updating of bandwidth parameters, overcomes the limitations of existing VMD methods in terms of parameter dependence and anti-aliasing performance, and achieves reliable, accurate and adaptive signal decomposition in complex environments.
[0034] Secondly, this application provides a variational mode decomposition device based on orthogonality criteria. The device includes: an empirical spectrum trend extraction module, used to perform background equalization on the input signal and extract the empirical spectrum trend; the empirical spectrum trend indicates the overall fluctuation of the target signal's spectral energy; the target signal includes mechanical vibration signals, biomedical signals, electromagnetic wave signals, acoustic wave signals, or artificial signals; an initial parameter setting module, used to determine initial parameters based on the empirical spectrum trend, the initial parameters including the number of decomposition modes, the initial center frequency of each mode, and the initial bandwidth; an orthogonal optimization module, used to construct an orthogonal constrained variational model based on the orthogonality of the input signal in the signal space, the orthogonality being reflected in the relationship between each mode and its corresponding residual; and a parameter update module, used to iteratively update each mode, the center frequency of each mode, and the bandwidth based on the orthogonal constrained variational model until the algorithm convergence condition is met, thereby obtaining the signal decomposition result.
[0035] Thirdly, this application provides a computing device, comprising: at least one memory for storing a program; and at least one processor for executing the program stored in the memory, wherein when the program stored in the memory is executed, the processor is configured to execute a method as described in any of the first aspects.
[0036] Fourthly, this application provides a computer-readable storage medium storing a computer program that, when run on a processor, causes the processor to perform the method as described in any of the first aspects. Attached Figure Description
[0037] To more clearly illustrate the technical solutions of the various embodiments disclosed in this specification, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only a few embodiments disclosed in this specification. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.
[0038] The accompanying drawings used in the description of the embodiments or prior art are briefly introduced below.
[0039] Figure 1 A typical research flowchart for optimizing initial parameters in VMD
[0040] Figure 2 This is a schematic diagram illustrating the filtering principle of VMD.
[0041] Figure 3 A flowchart of the variational mode decomposition method based on orthogonality criteria provided in the embodiments of this application;
[0042] Figure 4 A flowchart illustrating the method for extracting empirical spectrum trends provided in embodiments of this application;
[0043] Figure 5 This is a flowchart illustrating the method for setting initial parameters based on empirical spectrum trends in the embodiments of this application.
[0044] Figure 6 This is a schematic diagram of the orthogonal constrained variational model in the method provided in the embodiments of this application;
[0045] Figure 7 This is a schematic diagram illustrating the iterative update of modal parameters in the method provided in the embodiments of this application;
[0046] Figure 8 A schematic diagram comparing the modal decomposition performance of EST-AOCVMD, CEEMDAN, and VMD as provided in the embodiments of this application;
[0047] Figure 9 A schematic diagram showing the frequency resolution comparison of EST-AOCVMD, CEEMDAN, and VMD provided in the embodiments of this application;
[0048] Figure 10 A schematic diagram comparing the noise immunity performance of EST-AOCVMD, CEEMDAN, and VMD in a noisy environment, as provided in the embodiments of this application;
[0049] Figure 11 A schematic diagram comparing the computational efficiency and reconstruction accuracy of EST-AOCVMD, CEEMDAN, and VMD in embodiments of this application;
[0050] Figure 12 A schematic diagram of a variational mode decomposition device based on orthogonality criteria provided in an embodiment of this application;
[0051] Figure 13 A computing device provided in an embodiment of this application. Detailed Implementation
[0052] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the technical solutions in the embodiments of this application will be described below with reference to the accompanying drawings.
[0053] In the description of the embodiments of this application, the words "exemplary," "for example," or "for instance" are used to indicate examples, illustrations, or explanations. Any embodiment or design described as "exemplary," "for example," or "for instance" in the embodiments of this application should not be construed as being more preferred or advantageous than other embodiments or designs. Specifically, the use of the words "exemplary," "for example," or "for instance" is intended to present the relevant concepts in a specific manner.
[0054] In the description of the embodiments in this application, the term "and / or" is merely a description of the association relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A existing alone, B existing alone, and A and B existing simultaneously. Furthermore, unless otherwise stated, the term "multiple" means two or more. For example, multiple systems refer to two or more systems, and multiple terminals refer to two or more terminals.
[0055] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature. The terms "comprising," "including," "having," and their variations all mean "including but not limited to," unless otherwise specifically emphasized.
[0056] In the description of the embodiments in this application, "some embodiments" are mentioned, which describe a subset of all possible embodiments. However, it is understood that "some embodiments" can be the same subset or different subsets of all possible embodiments, and can be combined with each other without conflict.
[0057] In the description of the embodiments of this application, the terms "first, second, third, etc." or module A, module B, module C, etc. are used only to distinguish similar objects and do not represent a specific ordering of objects. It is understood that, where permitted, a specific order or sequence can be interchanged so that the embodiments of this application described herein can be implemented in an order other than that illustrated or described herein.
[0058] In the description of the embodiments of this application, the reference numerals for the steps, such as S110, S120, etc., do not necessarily indicate that the steps will be executed in this manner. Where permissible, the order of the steps can be interchanged or executed simultaneously.
[0059] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs. The terminology used herein is for the purpose of describing embodiments of this application only and is not intended to limit this application.
[0060] Figure 1 This is a flowchart illustrating the overall research process for optimizing initial parameters in VMD. Figure 1 As shown, for the preset parameters of VMD and its extended models, including the number of decomposed modes, bandwidth balance parameters and initial center frequency, the optimization algorithm is considered from four perspectives: objective function, optimization algorithm, iterative decomposition and initial center frequency estimation, so as to obtain the parameter input model and obtain the decomposition result.
[0061] The core idea of VMD is to extend the classical Wiener filter to multiple adaptive frequency bands and estimate all signal components simultaneously through a joint optimization method. Therefore, it can effectively avoid the error propagation problem in the component iterative extraction process and obtain the component estimation results.
[0062] The principle of VMD mainly includes the following steps: (1) Calculating the analytic signal of each signal component through Hilbert transform; (2) Shifting the signal component to baseband by mixing the analytic signal with a specific frequency exponential term; (3) Utilizing... Gaussian smoothing index is used to evaluate the signal bandwidth after the shift.
[0063] First, establish the following constrained variational optimization model:
[0064]
[0065] formula middle, For input signals; Represents the Dirac function; Represents convolution operation; The preset number of decomposition modes; For the mode to be estimated, The modal center frequency is to be estimated.
[0066] The variational constrained model can be transformed into an unconstrained variational model by introducing a quadratic penalty term and a Lagrange multiplier term:
[0067]
[0068] formula In the middle, the secondary penalty factor Used to balance the bandwidth of the mode with the fidelity of signal reconstruction; Represents the Lagrange multipliers; This represents the inner product operation. The quadratic penalty factor can be used. Let this be denoted as the bandwidth constraint parameter. For the formula... Solving this problem is equivalent to finding an unconstrained variational model. The saddle point. The formula can be modified using the alternating direction multiplier method. Iterative decomposition of unconstrained variational models.
[0069] In the In the nth iteration, the 1st The mode, center frequency, and Lagrange multipliers are updated in the frequency domain as follows:
[0070]
[0071]
[0072]
[0073] formula middle, , , Represent , as well as Fourier transform; is the step size parameter of the Lagrange multiplier.
[0074] The iteration termination condition for the above update process is as follows:
[0075]
[0076] formula middle These are the convergence parameters.
[0077] Through formula The sum of the variances of all modes in two consecutive iterations is used as the algorithm convergence evaluation index. When the algorithm convergence evaluation index is less than the preset convergence parameter... The iteration terminates under certain circumstances.
[0078] Ultimately, the time-domain waveforms of each mode can be obtained by applying the formula. The spectrum shown is obtained by performing an inverse Fourier transform. The modes extracted by VMD exhibit specific sparsity, quasi-orthogonality, and narrowband characteristics.
[0079] Figure 2This is a schematic diagram of the VMD filtering principle. VMD is essentially a Wiener filter bank with an adaptive center frequency. Figure 2 As shown, the number of decomposed modes Setting the value is a crucial step in VMD, and its value directly affects the decomposition effect of the signal: if the value is set too small, it will lead to under-decomposition and fail to fully extract the target component in the signal; conversely, if the value is set too large, it will cause over-decomposition, resulting in false modes or mode repetition.
[0080] Traditional methods for determining this key parameter typically rely on manual experience or exhaustive searches based on quantitative evaluation metrics. This is not only time-inefficient, but also problematic because the effectiveness of VMD is simultaneously affected by the number of modes. With bandwidth constraint parameters Due to the influence of the parameter, if only one parameter is optimized while the other parameter is set improperly, it is still difficult to obtain the ideal decomposition result.
[0081] In some possible implementations, intelligent optimization algorithms such as multi-objective particle swarm optimization, artificial fish swarm optimization, and whale optimization can be used, combined with specific fitness functions such as envelope entropy, sample entropy, and permutation entropy, to automatically find the number of decomposition modes that achieves the optimal decomposition effect. and bandwidth constraint parameters .
[0082] However, the above methods do not effectively solve the fundamental problem. VMD still has significant limitations in practical engineering applications under complex and variable environments and low signal-to-noise ratio conditions: On the one hand, intelligent optimization algorithms select parameters through iterative updates of the population, which brings a large amount of computation. On the other hand, an inappropriate objective function cannot guide global optimization and may get stuck in local optima without obtaining the global optimum.
[0083] The initialization method of modal center frequencies directly affects the decomposition results of VMD. In complex application scenarios, choosing an appropriate center frequency initialization method can yield ideal decomposition results.
[0084] Furthermore, if the center frequency of the potential components in the signal can be effectively estimated in advance, the efficiency of the algorithm can be significantly improved while determining the number of decomposition modes.
[0085] Currently, optimization based on modal initial center frequency includes: optimizing the center frequency according to the convergence trend of the algorithm's center frequency; or optimizing the center frequency based on the spectral morphology information of the input signal.
[0086] The former often requires multiple executions of VMD, and then the optimal center frequency of each mode is determined by analyzing the convergence trend under different initial center frequency values.
[0087] Existing spectral morphology methods typically use the peak frequency directly or divide the frequency bands by local minima and take the peak frequency of each sub-band as the initial center frequency, which has poor robustness in noisy environments.
[0088] Although VMD alleviates the mode aliasing phenomenon in the traditional EMD method to some extent by introducing bandwidth constraints and center frequency optimization mechanisms, its anti-aliasing ability still depends on the reasonable setting of model parameters.
[0089] When a signal contains components with highly similar frequencies or significantly different amplitudes, mode aliasing may still occur in VMD if the parameters are not selected properly, leading to distorted decomposition results. This indicates that the anti-aliasing capability of VMD in complex signal or high-noise scenarios still needs improvement.
[0090] In summary, based on the orthogonality criterion, this application proposes empirical spectral trend-guided adaptive orthogonal-constrained variational mode decomposition (EST-AOCVMD).
[0091] The variational mode decomposition method based on orthogonality criteria provided in this application is described in detail below with reference to the accompanying drawings and specific embodiments.
[0092] Figure 3 A flowchart illustrating the variational mode decomposition method based on orthogonality criteria provided in this application embodiment. Figure 3 As shown, it includes the following steps S31-S34.
[0093] S31 performs background equalization on the input signal and extracts the empirical spectrum trend; the empirical spectrum trend indicates the overall fluctuation of the target signal's spectral energy; the target signal includes mechanical vibration signals, biomedical signals, electromagnetic wave signals, sound wave signals, or artificial signals.
[0094] S32, the initial parameters are determined based on the empirical spectrum trend. The initial parameters include the number of decomposed modes, the initial center frequency of each mode, and the initial bandwidth.
[0095] S33. Construct an orthogonal constraint variational model based on the orthogonality of the input signal in the signal space. The orthogonality is reflected in the relationship between each mode and its corresponding residual.
[0096] S34, based on the orthogonal constraint variational model, iteratively updates each mode, the center frequency and bandwidth of each mode until the algorithm convergence condition is met, and obtains the signal decomposition result.
[0097] The variational mode decomposition method based on orthogonality criterion provided in this application adaptively determines the number of decomposed modes and the initial center frequency of the modes. Based on the orthogonality criterion of modes, it realizes dynamic updating of bandwidth parameters, overcomes the limitations of existing VMD methods in terms of parameter dependence and anti-aliasing performance, and achieves reliable, accurate and adaptive signal decomposition in complex environments.
[0098] The variational mode decomposition method based on orthogonality criteria provided in this application can be widely applied to technical scenarios such as mechanical vibration signal analysis, underwater acoustic signal feature extraction, weak signal detection and noise suppression.
[0099] The above steps will be further explained below with reference to the accompanying drawings and specific embodiments.
[0100] For example, Figure 4 A flowchart illustrating the method for extracting empirical spectrum trends provided in embodiments of this application. For example... Figure 4 As shown, step S31, which involves background equalization of the input signal and extraction of empirical spectrum trends, specifically includes the following steps S41-S43.
[0101] S41, the discrete signal sampled from the input digital system is subjected to a centrally symmetric mirror extension, and the extended time series is subjected to a discrete Fourier transform to retain the spectral components corresponding to its non-negative frequencies, thus obtaining a single-sided spectrum.
[0102] Time series of input signals Perform a centrally symmetric mirror continuation to obtain the continuation time series. For this extended time series Perform a discrete Fourier transform and retain the spectral components corresponding to the non-negative frequencies to obtain the one-sided spectrum. .
[0103] S42, based on this one-sided spectrum, the mean background noise is estimated by a threshold segmentation method and a median filtering method.
[0104] Based on the frequency sequence of the single-sided amplitude spectrum Calculate the mean background noise Specifically, in this embodiment 1, the mean background noise is calculated through the following steps S421-S425:
[0105] S421, in amplitude spectrum Both sides are respectively length of The mirror extension yields the extended first frequency sequence:
[0106]
[0107] S422, constructed with Centered on, with a length of sliding window This is used to calculate the local sample mean of the first frequency sequence.
[0108] S423, for windows within The data are sorted in ascending order to obtain the second frequency sequence:
[0109]
[0110] in, For the window The minimum value among the data. For the window The maximum value among the data.
[0111] S424, according to the second frequency sequence The local sample mean was calculated to be:
[0112]
[0113] S425, Values within the window greater than the first threshold The data is excluded, according to the formula. Calculate the mean background noise:
[0114]
[0115] in, The truncation length is defined to satisfy: , ; This is the first empirical threshold factor, typically ranging from 2 to 10. However, in practical engineering applications, it needs to be adjusted based on the noise level and signal characteristics. It should be adjusted appropriately. The mean of the second frequency sequence; The first empirical threshold is used to determine the truncation length when estimating the global noise mean.
[0116] S43, utilizing Gaussian smoothing method, from the single-sided amplitude spectrum and background noise mean Extract the observed signal spectrum trend and the background noise spectrum trend, and calculate the difference between the observed signal spectrum trend and the background noise spectrum trend as the empirical spectrum trend. Specifically, by solving the following optimization problem, the output signal spectrum and empirical spectrum trend are obtained:
[0117]
[0118] in, Denotes the Euclidean norm; It is the identity matrix; It is a second-order difference matrix:
[0119]
[0120] Next, step S32 is executed to use the signal spectrum and empirical spectrum trends for the initial parameter setting of the variational model.
[0121] For example, Figure 5 This is a flowchart illustrating the method for determining initial parameters based on empirical spectrum trends in an embodiment of this application. For example... Figure 5 As shown, S32 specifically includes the following steps S51-S54.
[0122] S51, utilizing empirical spectrum trends The local minimum points and the boundary frequencies of the spectrum determine the sub-band boundaries, dividing the frequency band into K sub-bands:
[0123]
[0124] Among them, the sub-band boundary frequency The sub-band boundary includes the zero frequency and the Nyquist sampling frequency. The zero frequency and the Nyquist frequency are considered as the first and second boundary frequencies of the entire band, i.e., the first boundary... Second boundary ; It is a natural number greater than or equal to 1.
[0125] S52, calculate the initial bandwidth constraint parameters of the mode based on the sub-band bandwidth.
[0126] First, the initial bandwidth of each mode is determined based on the boundary frequencies of each sub-band. Specifically, the first... Initial bandwidth of each sub-band The calculation is as follows:
[0127]
[0128] Then according to the first The initial bandwidth of each sub-band is determined. The initial bandwidth constraint parameters for each mode are calculated using the following formulas:
[0129]
[0130] S53, Traversal Each sub-band is used to determine the initial center frequency of each sub-band by utilizing the empirical spectral trend peak points and spectral peak points within the sub-band.
[0131] Specifically, for the first Sub-band peak point of empirical spectrum trend Corresponding experience threshold The second threshold, when the second threshold Greater than the peak value of the amplitude spectrum within the sub-band At that time, the first The center frequency of each signal component is determined as the frequency corresponding to the peak point of the empirical spectrum trend. Conversely, when the second threshold Less than or equal to the peak value of the amplitude spectrum within the sub-band At that time, the first The center frequency of each sub-band is determined as the frequency corresponding to the peak point of the spectrum. .
[0132] in, This is the second empirical threshold factor, typically ranging from 2 to 10, and its value can be the same as the first empirical threshold factor mentioned in the preceding steps. While maintaining consistency is important, in practical engineering applications, adjustments need to be made based on noise levels and signal characteristics. It should be adjusted appropriately; This represents the input variable value that maximizes the output of the objective function.
[0133] Thus, the method provided in this application determines the number of sub-bands based on the empirical spectrum trend by performing background equalization and empirical spectrum trend extraction on the input signal. The boundary frequency and center frequency of each sub-band are used to achieve adaptive frequency band division and center frequency estimation of signal components.
[0134] S54, determine the number of decomposed modes, the initial center frequency of each mode, and the initial bandwidth.
[0135] Specifically, it can be based on the number of sub-bands. The number of decomposed modes is determined by the center frequency corresponding to each sub-band. and The initial center frequency of each mode, according to The boundary frequencies of each sub-band determine the initial bandwidth of the mode.
[0136] After setting the initial parameters of the variational model, step S33 is executed to construct an orthogonal constrained variational model based on the orthogonality of the input signal in the signal space.
[0137] For example, Figure 6 This is a schematic diagram of the orthogonal constrained variational model in the method provided in the embodiments of this application. Figure 6 As shown, step S33 can determine the orthogonal constraint variational model through the following steps S61-S63.
[0138] S61, Suppose the input signal is orthogonally projected into the signal space, and the orthogonal projection includes... One modality and one residual term.
[0139] Based on the number of sub-bands Decompose the input signal into One modality and one residual term:
[0140]
[0141] in, Input signal In terms of modality The orthogonal projection onto a specific basis in the spanned signal space then gives the orthogonality condition for modal decomposition of the input signal:
[0142]
[0143] formula In the middle, the first The residuals corresponding to each mode for:
[0144]
[0145] According to the formula It can be seen that each mode occupies a major component in its corresponding signal subspace. The orthogonality condition is used to ensure that the spectral overlap between each mode and its corresponding residual is minimized, which can... and This is denoted as an orthogonal constraint term.
[0146] S62, orthogonality is manifested between each mode and its corresponding residual. According to Planck's theorem, the orthogonality between a mode and its corresponding residual is expressed in the frequency domain as:
[0147]
[0148] formula middle, represent Fourier transform; Represents complex conjugation.
[0149] S63, based on the orthogonal constraint conditions and the formula The new orthogonal constrained variational model is constructed as follows:
[0150]
[0151] formula middle, For the first The frequency response of a filter with residuals corresponding to each mode is expressed as:
[0152]
[0153] Among them, the Effective bandwidth of each mode as follows:
[0154]
[0155] For a smooth transition function:
[0156]
[0157] Therefore, in the variational mode decomposition method based on the orthogonality criterion provided in Embodiment 1 of this application, constraints based on the orthogonality criterion are introduced, and an empirical wavelet filter is constructed to constrain the residual term, which can directly and effectively suppress mode aliasing and has physical interpretability.
[0158] Furthermore, S34 is executed based on the orthogonal constraint variational model. The spectrum, center frequency, and bandwidth of each mode are iteratively updated.
[0159] For example, Figure 7 This is a schematic diagram illustrating the iterative update of modal parameters in the method provided in this application embodiment. For example... Figure 7 As shown, S34 includes:
[0160] S71, using the Lagrange multiplier method to solve the optimization problem of an orthogonally constrained variational model, formula. The augmented Lagrange equation can be expressed as:
[0161]
[0162] S72, the above augmented Lagrange equation is solved iteratively in the frequency domain using the alternating direction multiplier method.
[0163] S721, specifically, in the... In the nth iteration, the 1st The sub-model of each modality is updated in the frequency domain as follows:
[0164]
[0165] S722, correspondingly, the first The center frequency corresponding to each mode is updated in the frequency domain as follows:
[0166]
[0167] formula This indicates that the method provided in the embodiments of this application can calculate the modal power spectrum. The center frequency is estimated by the centroid. .
[0168] Furthermore, in the early stages of optimization, a wider filter bandwidth helps to quickly estimate the approximate frequency band of the target component; while as the iterative solution progresses and the modes gradually converge to the desired frequency band, a narrower bandwidth is beneficial for suppressing noise and other interference.
[0169] Therefore, in the optimization process, the method provided in the application embodiment is based on the orthogonality criterion of modality and the matching pursuit algorithm, and designs an adaptive update mechanism for bandwidth constraint parameters from coarse to fine.
[0170] S723 employs a coarse-to-fine adaptive mechanism to update the initial bandwidth constraint parameters, combined with... With formula We can obtain:
[0171]
[0172] in, This indicates taking the real part of the complex number. The formula... Substitute into the above formula We can obtain:
[0173]
[0174] Solve the above equation using the fixed-point iteration method. , thus we can obtain the first The update formula for the bandwidth constraint parameters corresponding to each mode is:
[0175]
[0176] Furthermore, according to the formula Update # A filter that corresponds to the residual of each mode. .
[0177] S724, Lagrange multipliers updated as follows:
[0178]
[0179] S73, in the In the next iteration, when the sum of the variances of all modes in two adjacent iterations is less than the preset convergence parameter value:
[0180]
[0181] The algorithm reaches the convergence condition, terminates the iteration, and outputs the result. The signal decomposition results are obtained by analyzing each mode and its center frequency.
[0182] The variational mode decomposition method based on orthogonality criteria provided in this application adaptively adjusts the bandwidth constraint parameters of each mode according to the measure of orthogonality in the frequency domain during the iterative optimization process. This mechanism overcomes the limitations of traditional VMD where bandwidth is globally fixed or manually set, achieving a dynamic balance between decomposition accuracy and flexibility.
[0183] The results analysis and performance verification are as follows.
[0184] First, the filtering characteristics of the method provided in this application embodiment are evaluated using fractional Gaussian noise (FGN). The power spectral density (PSD) of FGN has a power-law relationship with frequency, and its variation trend is determined by the Hearst exponent. ( This index is used to measure the long-term dependence of time series. Specifically, when... When FGN degenerates into white Gaussian noise (WGN), there is no correlation between different times; when When FGN exhibits a negative correlation, that is, adjacent increments tend to change in opposite directions; when At that time, FGN showed a positive correlation, meaning that adjacent increments tended to change in the same direction.
[0185] The filtering characteristics of the method provided in the embodiments of this application were evaluated using Monte Carlo numerical simulation experiments: 5000 independent and repeated experiments were conducted on an FGN of length 2048, with a noise variance of 1 and the Hearst exponent set to... , and In each experiment, the EST-AOCVMD, complete ensemble EMD (CEEMDAN), and VMD provided in the embodiments of this application were applied to decompose the same noise data, and the average PSD of each mode was calculated. The results are as follows: Figure 8 As shown.
[0186] according to Figure 8 As can be seen from (a), (b), and (c), the EST-AOCVMD provided by the embodiments of this application exhibits more adaptive mode decomposition performance compared to CEEMDAN and VMD: when the Hearst exponent is The mode decomposition results of EST-AOCVMD show a positive correlation with the input noise data. Its frequency band division exhibits a multi-resolution characteristic similar to wavelet packet transform, with high resolution in the low-frequency part and low resolution in the high-frequency part.
[0187] When the Hearst exponent is Noise is negatively correlated, or When there is no correlation, the mode decomposition results of EST-AOCVMD in this application tend to be uniformly distributed compared with CEEMDAN and VMD, exhibiting constant bandwidth characteristics similar to short-time Fourier transform.
[0188] This phenomenon indicates that the EST-AOCVMD provided in this application embodiment can adaptively adjust the frequency domain decomposition strategy according to the inherent long-range correlation structure of the signal. Furthermore, according to... Figure 8 As can be seen from (a) and (b), the adjacent modes of CEEMDAN and VMD have significant overlap in the frequency domain, indicating that their mode separation capability is limited and they are prone to mode aliasing. In contrast, the mode spectrum of EST-AOCVMD provided in this application embodiment has almost no overlap, showing clearer frequency band division and stronger anti-mode aliasing capability.
[0189] Next, the frequency resolution of the EST-AOCVMD method provided in this application embodiment is evaluated using a classic "tone separation" experiment. The following synthesized signal model is constructed:
[0190]
[0191] formula middle, and These are the amplitudes of the two components; the frequency satisfies... Define the amplitude ratio. The relative error of decomposition is used as the performance evaluation index.
[0192] The EST-AOCVMD method provided in this application embodiment was applied to this signal and compared with CEEMDAN and Successive VMD (SVMD). The results are as follows: Figure 9 As shown.
[0193] like Figure 9 The dark color shown in (a) indicates that CEEMDAN has significant decomposition errors in multiple frequency bands. This is because its frequency depends on local extreme point sampling, and modal aliasing is unavoidable when the frequency of the signal components is close, resulting in the inability to effectively separate the tone components.
[0194] like Figure 9 Although SVMD shown in Figure (b) alleviates the mode aliasing problem to some extent, it is prone to mode repetition, resulting in a moderate degree of relative decomposition error in a large number of regions.
[0195] In comparison, such as Figure 9As shown in (c), the EST-AOCVMD achieves accurate tone separation across almost the entire frequency domain, with performance degradation only occurring near the Nyquist frequency.
[0196] Next, the performance of the method provided in this application embodiment under noisy environments will be evaluated using two typical background noise scenarios: WGN and colored Gaussian noise (CGN). CGN will be modeled using a simplified model of ship radiated noise, with a power spectrum peaking at 400 Hz and attenuating towards both sides at a slope of −6 dB / octave. The experiment will use the following multi-component noisy signal model:
[0197]
[0198] formula The intermediate parameter is set to: FM component center frequency ,bandwidth Signal duration Sampling frequency: 2 kHz; Additive noise It follows a normal distribution with a mean of 0 and a variance of 1. , , These represent the amplitudes of the AM, CW, and FM components, respectively. , , , These represent the initial phases of the AM component amplitude modulation term, AM component carrier, CW component, and FM component, respectively, all of which are independent of each other and follow an interval... A uniform distribution on the surface.
[0199] In the experiment, only one component was considered as the target signal each time, while the other two components were considered as interference signals, with their amplitudes set to twice that of the target signal. The signal-to-noise ratio (SNR) was controlled by adjusting the amplitude of the target component, and the correlation coefficient (CC) was used as a quantification index: the CC value between the extracted mode and the clean signal of the corresponding target component was calculated. The closer the CC value is to 1, the better the noise robustness, the better the denoising effect, and the stronger the signal feature extraction capability. The EST-AOCVMD method provided in this application embodiment was compared with several representative noise reduction methods: CEEMDAN, VMD, SVMD, Adaptive Energy-Constrained VMD (AECVMD), and Bandwidth-Aware Adaptive Chirp Mode Decomposition (BA-ACMD) under the same conditions, with each method running 1000 Monte Carlo simulations independently.
[0200] The results are as follows Figure 10 As shown in the figure, the solid line (CGN) and dashed line (WGN) represent the baseline CC curves between the noisy signal and the noiseless component, respectively. It can be seen that EST-AOCVMD achieves the highest CC values under both types of noise backgrounds, with its advantage being particularly significant under low signal-to-noise ratio (high noise) conditions. In contrast, other methods struggle to robustly extract the target component and achieve effective denoising, and their applicability is limited to specific signal types or noise conditions, especially with severe performance degradation in CGN scenarios. Specifically, when the CW component is used as the target signal, its frequency is close to the 400 Hz spectral peak of CGN, posing a significant challenge to most methods. However, the EST-AOCVMD provided in this application embodiment can still accurately estimate its center frequency and effectively extract the target component, not only verifying its good adaptability to broadband signals but also highlighting its significant advantage in extracting narrowband signal components under complex noise backgrounds.
[0201] To further analyze the practical engineering application value of the proposed method, its computational efficiency and reconstruction accuracy for processing multi-component signals are quantitatively evaluated and compared with other classic AMD methods. The experiment uses the following noisy harmonic signal model:
[0202]
[0203] in, The input signal-to-noise ratio is zero-mean white Gaussian noise with its variance adjusted to be constant at 10 dB; the signal duration is 1 second and the sampling frequency is set to 4096 Hz.
[0204] Under the same hardware and software environment (Intel Core i5-12400F @ 2.5 GHz, 16 GB DDR4 memory, Windows 11 operating system, MATLAB R2022b), each method was run independently for 1000 Monte Carlo simulations. The average execution time (ET) was used as the computational efficiency indicator, and the reconstruction accuracy was evaluated by the CC between the reconstructed signal and the original clean signal.
[0205] The results are as follows Figure 11 As shown. The results show that the EST-AOCVMD provided in this application achieves the highest reconstruction accuracy with the lowest or near-lowest computational overhead in almost all test scenarios. This advantage mainly stems from its empirical spectrum trend-guided initial parameter settings and adaptive orthogonal constraint mechanism, which significantly accelerates the convergence of the variational optimization process. Although when the number of modes... At the same time, the execution time of the EST-AOCVMD algorithm provided in this application embodiment is slightly higher than that of BA-ACMD, but both are still far superior to other comparative methods such as CEEMDAN, VMD, and SVMD, showing significant computational efficiency advantages.
[0206] In summary, addressing the key issues of existing mode decomposition methods such as blind parameter initialization, mode aliasing, lack of parameter adaptability, and insufficient noise robustness, the EST-AOCVMD method provided in this application accurately initializes model parameters through empirical spectral trends and introduces new constraint terms and bandwidth adaptive update mechanisms into the variational model based on the mode orthogonality criterion. This not only possesses excellent anti-mode aliasing capabilities and higher frequency resolution but also exhibits significant noise robustness and denoising performance in complex noise environments.
[0207] The EST-AOCVMD method provided in this application provides reliable support for multi-component signal analysis and feature extraction, while also having good computational efficiency, making it particularly suitable for real-time processing scenarios of multi-component non-stationary signals.
[0208] This application embodiment constructs a multi-component signal containing AM, CW, and FM components. In the experiment, one of the components is set as the target signal, and the remaining components are regarded as interference signals. Simulation experiments in various challenging scenarios with strong colored noise interference and dense frequency components comprehensively evaluate the signal extraction capability of the method. The system verifies the comprehensive advantages of the EST-AOCVMD method provided by this application embodiment in terms of noise robustness, denoising capability, and computational efficiency.
[0209] Experiments show that, compared with existing AMD methods such as CEEMDAN, SVMD, and BA-ACMD, the EST-AOCVMD method provided in this application embodiment can still effectively extract target components in complex environments and has good real-time processing potential, fully demonstrating its engineering practical value in fields such as mechanical fault diagnosis, biomedical signal analysis, and radar / sonar signal feature extraction.
[0210] The above is an introduction to the variational mode decomposition method based on orthogonality criteria provided in the embodiments of this application. It is understood that the sequence number of each step in the above embodiments does not imply the order of execution. The execution order of each process should be determined by its function and internal logic, and should not constitute any limitation on the implementation process of the embodiments of this application. Furthermore, in some possible implementations, each step in the above embodiments may be selectively executed according to actual conditions; it may be partially or fully executed, without limitation here. In addition, all or part of any feature of any of the above embodiments can be freely and arbitrarily combined without contradiction; the combined technical solution is also within the scope of this application.
[0211] Next, based on the above, the variational mode decomposition device based on the orthogonality criterion provided in the embodiments of this application will be introduced. For relevant descriptions of concepts, formulas, etc., involved in the following content, please refer to the above text.
[0212] Figure 12 This application provides a variational mode decomposition device based on orthogonality criteria. For example... Figure 12 As shown, the variational mode decomposition device 120 based on orthogonality criteria includes an empirical spectrum trend extraction module 121, an initial parameter setting module 122, an orthogonal optimization module 123, and a parameter update module 124.
[0213] The empirical spectrum trend extraction module 121 performs background equalization on the input signal and extracts the empirical spectrum trend; the empirical spectrum trend indicates the overall fluctuation of the target signal's spectral energy; the target signal includes mechanical vibration signal, biomedical signal, electromagnetic wave signal, sound wave signal, or artificial signal.
[0214] The initial parameter setting module 122 determines the initial parameters based on the empirical spectrum trend. The initial parameters include the number of decomposed modes, the initial center frequency of each mode, and the initial bandwidth.
[0215] The orthogonal optimization module 123 constructs an orthogonal constrained variational model based on the orthogonality of the input signal in the signal space. The orthogonality is reflected between each mode and its corresponding residual.
[0216] The parameter update module 124 iteratively updates each mode, the center frequency and bandwidth of each mode based on the orthogonal constraint variational model until the algorithm convergence condition is met, and obtains the signal decomposition result.
[0217] In the device, a module, as an example of a software functional unit, may include code running on a computing instance, such as the experience spectrum trend extraction module 121. The computing instance may include at least one of a physical host (computing device), a virtual machine, or a container. Further, the aforementioned computing instance may be one or more. For example, the experience spectrum trend extraction module 121 may include code running on multiple hosts / virtual machines / containers. It should be noted that the multiple hosts / virtual machines / containers used to run the code may be distributed in the same region or in different regions. Further, the multiple hosts / virtual machines / containers used to run the code may be distributed in the same availability zone (AZ) or in different AZs, each AZ including one or more geographically proximate data centers. Typically, a region may include multiple AZs.
[0218] Similarly, multiple hosts / virtual machines / containers used to run this code can be distributed within the same Virtual Private Cloud (VPC) or across multiple VPCs. Typically, a VPC is set up within a region. Communication between two VPCs within the same region, as well as between VPCs in different regions, requires a communication gateway to be set up within each VPC to enable interconnection between VPCs.
[0219] This application also provides a computing device 130. For example... Figure 13 As shown, the computing device 130 includes a bus 132, a processor 134, a memory 136, and a communication interface 138. The processor 134, the memory 136, and the communication interface 138 communicate with each other via the bus 132. The computing device 130 can be a computing device or a terminal device. It should be understood that this application does not limit the number of processors and memories in the computing device 130.
[0220] Bus 132 can be a Peripheral Component Interconnect (PCI) bus or an Extended Industry Standard Architecture (EISA) bus, etc. Buses can be categorized as address buses, data buses, control buses, etc. For ease of representation, Figure 13The bus 134 is represented by a single line, but this does not mean that there is only one bus or one type of bus. The bus 134 may include a path for transmitting information between various components of the computing device 130 (e.g., memory 136, processor 134, communication interface 138).
[0221] Processor 134 may include any one or more processors such as a central processing unit (CPU), a graphics processing unit (GPU), a microprocessor (MP), or a digital signal processor (DSP).
[0222] Memory 136 may include volatile memory, such as random access memory (RAM). Processor 104 may also include non-volatile memory, such as read-only memory (ROM), flash memory, hard disk drive (HDD), or solid state drive (SSD).
[0223] The memory 136 stores executable program code, and the processor 134 executes the executable program code to implement the aforementioned functions. Figure 12 The empirical spectrum trend extraction module 121 shown herein performs the functions of all or part of the steps of the method in the above embodiments. That is, the memory 136 stores instructions for performing all or part of the steps of the method in the above embodiments.
[0224] The communication interface 138 uses transceiver modules such as, but not limited to, network interface cards and transceivers to enable communication between the computing device 130 and other devices or communication networks.
[0225] This application provides a computer-readable storage medium storing a computer program that, when run on a processor, causes the processor to perform the method described in any of the above embodiments.
[0226] It is understood that the processor in the embodiments of this application may be a central processing unit (CPU), or other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, transistor logic devices, hardware components, or any combination thereof. A general-purpose processor may be a microprocessor or any conventional processor.
[0227] The method steps in the embodiments of this application can be implemented in hardware or by a processor executing software instructions. The software instructions can consist of corresponding software modules, which can be stored in random access memory (RAM), flash memory, read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), registers, hard disks, portable hard disks, CD-ROMs, or any other form of storage medium known in the art. An exemplary storage medium is coupled to the processor, enabling the processor to read information from and write information to the storage medium. Of course, the storage medium can also be a component of the processor. The processor and the storage medium can reside in an ASIC.
[0228] In the above embodiments, implementation can be achieved, in whole or in part, through software, hardware, firmware, or any combination thereof. When implemented in software, it can be implemented, in whole or in part, as a computer program product. A computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, all or part of the flow or function according to the embodiments of this application is generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in or transmitted through a computer-readable storage medium. The computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired (e.g., coaxial cable, fiber optic, digital subscriber line (DSL)) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium that a computer can access or a data storage device such as a server or data center that integrates one or more available media. The available medium can be a magnetic medium (e.g., floppy disk, hard disk, magnetic tape), an optical medium (e.g., DVD), or a semiconductor medium (e.g., solid-state disk (SSD)).
[0229] It is understood that the various numerical designations used in the embodiments of this application are merely for descriptive convenience and are not intended to limit the scope of the embodiments of this application.
Claims
1. A variational mode decomposition method based on orthogonality criteria, applied to multi-component non-stationary signal processing in complex noise environments, characterized in that... The method includes: Background equalization is performed on the input signal to extract the empirical spectrum trend; the empirical spectrum trend indicates the overall fluctuation of the target signal's spectral energy; the target signal includes mechanical vibration signals, biomedical signals, electromagnetic wave signals, sound wave signals, or artificial signals; Initial parameters are determined based on the empirical spectrum trend, including the number of decomposed modes, the initial center frequency of each mode, and the initial bandwidth. An orthogonal constraint variational model is constructed based on the orthogonality of the input signal in the signal space, wherein the orthogonality is reflected between each mode and its corresponding residual; Based on the orthogonal constraint variational model, the center frequency and bandwidth of each mode are iteratively updated to obtain the signal decomposition results.
2. The method according to claim 1, characterized in that, The orthogonally constrained variational model includes: in, For the mode to be estimated, Let be the center frequency of the mode to be estimated; This is a modal bandwidth constraint parameter used to balance the bandwidth of the mode with the fidelity of signal reconstruction; Input signal In terms of modality The orthogonal projection onto a specific basis in the spanned signal space; For residual terms; For the first The residuals corresponding to each mode; For the first A filter that corresponds to the residual of each mode.
3. The method according to claim 1 or 2, characterized in that, The method further includes: solving the optimization problem of the orthogonal constrained variational model using the Lagrange multiplier method, to obtain the augmented Lagrange equation of the orthogonal constrained variational model as follows: in, Represents the Lagrange multipliers; This represents inner product operations.
4. The method according to claim 1 or 2, characterized in that, The iterative update of each mode, the center frequency and bandwidth of each mode based on the orthogonal constrained variational model includes: decomposing the optimization problem of the orthogonal constrained variational model into a series of sub-optimization problems using the alternating direction multiplier method, and solving them iteratively in the frequency domain; In the In the nth iteration, the th Each mode and its center frequency are updated in the frequency domain as follows: in, , , Represent , as well as Fourier transform.
5. The method according to claim 1 or 2, characterized in that, The iterative update of each mode, its center frequency, and its bandwidth based on the orthogonal constrained variational model includes: according to the... The initial bandwidth of each sub-band is determined. The initial bandwidth constraint parameters for each mode are updated using a coarse-to-fine adaptive mechanism. In the nth iteration, the 1st The bandwidth constraint parameters corresponding to each mode are: in, , They are respectively , Fourier transform; This indicates taking the real part of a complex number.
6. The method according to claim 3, characterized in that, The iterative update of each mode, its center frequency, and its bandwidth based on the orthogonal constrained variational model includes: in the first... In the next iteration, the Lagrange multipliers are updated as follows: in, for Fourier transform; Let be the step size parameter of the Lagrange multiplier.
7. The method according to claim 2, 3, 4 or 6, characterized in that, The first A filter that corresponds to the residual of each mode. The frequency response is: in, For a smooth transition function; the first Effective bandwidth of each mode for: 。 8. The method according to any one of claims 4-6, characterized in that, The iterative update of each mode, its center frequency, and its bandwidth based on the orthogonal constrained variational model includes: The sum of the variances of all modes in two consecutive iterations is used as the convergence evaluation index. When the convergence evaluation index is less than the preset convergence parameter... Under the following conditions, the algorithm convergence condition is met: If the algorithm convergence condition is met, the process is terminated. Iterative updates of the spectrum, center frequency, and bandwidth of each mode.
9. A variational mode decomposition device based on the orthogonality criterion, characterized in that, The device includes: The empirical spectrum trend extraction module is used to perform background equalization on the input signal and extract the empirical spectrum trend; the empirical spectrum trend indicates the overall fluctuation of the target signal's spectral energy; the target signal includes mechanical vibration signals, biomedical signals, electromagnetic wave signals, sound wave signals, or artificial signals; An initial parameter setting module is used to determine initial parameters based on the empirical spectrum trend. The initial parameters include the number of decomposed modes, the initial center frequency of each mode, and the initial bandwidth. The orthogonal optimization module is used to construct an orthogonal constrained variational model based on the orthogonality of the input signal in the signal space, wherein the orthogonality is reflected between each mode and its corresponding residual; The parameter update module is used to iteratively update each mode, the center frequency and bandwidth of each mode based on the orthogonal constraint variational model until the algorithm convergence condition is met, and obtain the signal decomposition result.
10. A computing device, characterized in that, include: At least one memory for storing programs; At least one processor is configured to execute a program stored in the memory, wherein when the program stored in the memory is executed, the processor is configured to perform the method as claimed in any one of claims 1-8.