VMD-CEEMDAN-driven IMU (Inertial Measurement Unit) adaptive noise reduction method

Abstract

The invention provides a VMD-CEEMDAN driven IMU (Inertial Measurement Unit) adaptive noise reduction method, and relates to the technical field of digital signal processing. According to the method, frequency domain analysis of variational mode decomposition and complete ensemble empirical mode decomposition can be combined, and a self-adaptive noise parameter optimization mechanism based on a signal inherent frequency domain structure is established. The optimal noise level is dynamically determined for each decomposition level by extracting the energy distribution, the frequency characteristic and the bandwidth characteristic of each mode, so that the parameter setting problem is fundamentally solved, the mode aliasing phenomenon is effectively avoided, and the signal decomposition quality and the noise reduction effect are remarkably improved.

Description

Technical Field

[0001] This invention relates to the field of digital signal processing technology, and to, but is not limited to, a VMD-CEEMDAN driven IMU adaptive noise reduction method, apparatus, and device. Background Technology

[0002] When unmanned aerial vehicle (UAV) swarms perform collaborative tasks in complex and dynamic environments, they face severe navigation and collaborative control challenges. In complex environments where satellite signals such as GPS are blocked, communication links are unstable, or electromagnetic interference is severe, traditional external systems relying on satellite positioning often fail to provide reliable navigation information. In these situations, inertial measurement units (IMUs) and other inertial sensors become crucial technologies for maintaining the autonomous navigation capabilities of UAV swarms. In research on path prediction and conflict resolution for UAV swarms, multi-sensor data fusion strategies are widely used to improve the comprehensiveness of environmental perception and data stability. However, a single IMU sensor is susceptible to multi-source noise from vibration, temperature changes, and electromagnetic interference in complex and dynamic environments, leading to increased uncertainty and significantly reduced robustness of its measurement data. This data quality deficiency directly degrades the trajectory prediction accuracy and collaborative control efficiency of the UAV swarm, thus becoming a key bottleneck restricting the safety and mission completion rate of the swarm system.

[0003] Existing IMU signal processing techniques have significant limitations when dealing with complex environmental noise. While the traditional Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) method improves upon Empirical Mode Decomposition (EMD) algorithms in suppressing mode aliasing, its core limitation lies in its reliance on fixed empirical parameter settings for noise addition strategies. This non-fixed limitation is particularly pronounced when processing complex, non-stationary IMU signals generated during the high-dynamic flight of UAVs. Because it cannot adaptively adjust based on the actual frequency domain characteristics of the signal, this type of method is highly susceptible to mode aliasing under different flight states, environmental conditions, and interference intensities, severely impacting signal decomposition quality and subsequent navigation calculation accuracy.

[0004] Furthermore, existing signal processing algorithms generally suffer from high computational complexity and insufficient real-time performance, making it difficult to meet the dual requirements of high-frequency data processing and real-time decision-making in UAV swarms. The traditional CEEMDAN method requires numerous ensemble averaging operations, with its computational complexity increasing exponentially with signal length and the number of decomposition layers. In dense collaborative operation scenarios involving UAV swarms, a large amount of IMU data needs to be processed in real time to support rapid path planning and dynamic conflict resolution. The efficiency bottleneck of traditional methods severely restricts the overall response speed and collaborative performance of the swarm system.

[0005] Meanwhile, the termination conditions set by existing algorithms lack scientific rigor and adaptability. Most methods rely solely on simple iteration count limits or fixed energy thresholds for judgment, failing to make precise termination decisions based on the actual decomposition state and frequency domain characteristics of the signal. This coarse stopping strategy easily leads to over-decomposition or under-decomposition problems, affecting not only the signal reconstruction accuracy but also potentially introducing additional computational burden and processing errors. Summary of the Invention

[0006] Based on the problems in related technologies, this invention provides a VMD-CEEMDAN-driven IMU adaptive noise reduction method, apparatus, and device to solve key technical problems in the prior art, such as noise parameter fixation, low computational efficiency, and unreasonable decomposition termination conditions, thereby improving the navigation accuracy and collaborative efficiency of UAV swarms in complex environments.

[0007] The technical solution of this invention is implemented as follows: This invention provides a VMD-CEEMDAN-driven IMU adaptive noise reduction method, the method comprising: The raw IMU signals acquired by the IMU sensor are sequentially standardized to obtain standardized IMU signals; The variational mode decomposition algorithm is used to adaptively decompose the standardized IMU signal in the frequency domain to obtain the variational mode decomposition result; The energy distribution characteristics, average frequency parameters, bandwidth characteristics, and energy time-varying properties of each mode are extracted from the variational mode decomposition results. Based on the energy distribution characteristics, the average frequency parameter, the bandwidth characteristics, and the time-varying energy properties, a precise mapping relationship from the inherent frequency domain structure of the signal to the noise intensity parameter is constructed; and by considering the modal energy weights, frequency position distribution, and bandwidth characteristics, the adaptive noise level is dynamically determined for each decomposition level of the adaptive noise complete set empirical mode decomposition. Using the aforementioned adaptive noise level, multiple sets of noise-assisted signals are constructed and ensemble averaging is performed to extract the intrinsic mode function components layer by layer. A comprehensive judgment mechanism based on residual signal Hilbert transform amplitude analysis and extreme point statistics is established; the decomposition process is automatically terminated when the residual signal meets the preset amplitude threshold condition or extreme point number threshold condition. The intrinsic mode function components and the final residual are linearly reconstructed to generate the reconstructed IMU signal.

[0008] This invention provides a VMD-CEEMDAN-driven IMU adaptive noise reduction device, the device comprising: The processing module is used to perform standardization processing on the raw IMU signals acquired by the IMU sensor in sequence to obtain standardized IMU signals; The decomposition module is used to perform adaptive frequency domain decomposition on the standardized IMU signal using a variational mode decomposition algorithm to obtain the variational mode decomposition result. The extraction module is used to extract the energy distribution characteristics, average frequency parameters, bandwidth characteristics, and energy time-varying characteristics of each mode from the variational mode decomposition results. The determination module is used to construct an accurate mapping relationship from the inherent frequency domain structure of the signal to the noise intensity parameter based on the energy distribution characteristics, the average frequency parameter, the bandwidth characteristics, and the time-varying energy characteristics; and to dynamically determine the adaptive noise level for each decomposition level of the adaptive noise complete set empirical mode decomposition by considering modal energy weights, frequency position distribution, and bandwidth characteristics. The extraction module is also used to extract the intrinsic mode function components layer by layer by constructing multiple sets of noise-assisted signals and performing ensemble averaging processing using the adaptive noise level. A module is established to create a comprehensive judgment mechanism based on residual signal Hilbert transform amplitude analysis and extreme point statistics; the decomposition process is automatically terminated when the residual signal meets the preset amplitude threshold condition or extreme point number threshold condition. The generation module is used to linearly reconstruct the intrinsic mode function components and the final residual to generate the reconstructed IMU signal.

[0009] In some embodiments, the processing module is further configured to, according to the formula For the original IMU signal The standardized IMU signal is obtained by performing standardization processing. ; In the formula, It is the standard deviation of the signal, defined as , This indicates the signal length, i.e., the number of sampling points; This represents the signal sampling point index, with a value ranging from 1 to... ; Indicates the first The signal value at each sampling point; It is the signal mean. .

[0010] In some embodiments, the extraction module is further configured to apply the VMD decomposition algorithm to the standardized IMU signal and solve the following variational problem to obtain the VMD decomposition result; The energy distribution characteristics, the average frequency parameters, the bandwidth characteristics, and the energy time-varying characteristics are extracted from the VMD decomposition results. The variational problem is: ; In the formula, The first value obtained from VMD decomposition is represented as... One modal component, This is the index of the modal component, with a value range of 1. , This indicates the number of VMD modes, representing how many modal components the signal is decomposed into; Indicates the first The center frequency of each mode, superscript The center indicates the central position of the mode in the frequency domain; This represents the Dirac impulse function, used to construct analytic signals; This represents the time partial derivative operator, used to calculate the time derivative of a signal; Indicates the convolution operation; Represents the imaginary unit; express The square of the norm; t represents the time variable, corresponding to the sampling time in discrete implementation; The phrase "bound to" refers to the constraints of a variational problem. The VMD decomposition result is as follows: ; In the formula, This represents the bandwidth constraint parameter, used to control the bandwidth of VMD decomposition; the larger the value, the smaller the bandwidth. This represents the noise tolerance parameter, which controls the VMD's sensitivity to noise. The larger the value, the less sensitive it is to noise. This indicates whether a DC component is included, with a value of 0 (not included) or 1 (included). This represents the convergence tolerance, controls the VMD iteration termination condition, and represents the relative error threshold. The energy distribution characteristics Represented as: ; In the formula, This represents the energy of the p-th mode, reflecting the importance of that mode in the signal; This represents the amplitude of the p-th mode at the j-th sampling point; The average frequency parameter Represented as: ; In the formula, Indicates the first The average frequency of each mode is expressed in Hertz (Hz). Indicates the first The mode in the th ... The instantaneous frequency at each sampling point; Used to convert angular frequency to Hertz frequency; The bandwidth characteristics Represented as: ; In the formula, Indicates the first The bandwidth of each mode reflects the width of the frequency distribution; Indicates the first The mode in the th ... The instantaneous frequency at each sampling point; This represents the mean of instantaneous frequencies. The overline indicates the averaging operation, defined as: ; The energy time-varying characteristics Represented as: ; In the formula, Indicates the first The energy change rate of each mode reflects the degree of energy change in the time domain; This represents the size of the moving average window, typically taken as... ; Indicates the first Each modality in Energy within a local window centered on the superscript To represent a local area, it is defined as: , This represents the index of the sampling point within the window, with a value range of [value range missing]. arrive ; Indicates the first The mode in the th ... Amplitude at each sampling point; local energy mean Defined as: ; Indicates the first The average energy of all local windows for each mode; This represents the total number of windows.

[0011] In some embodiments, the construction module is further configured to normalize the energy distribution characteristics, the bandwidth characteristics, and the energy time-varying characteristics respectively to obtain normalized energy distribution characteristics, normalized bandwidth characteristics, and normalized energy time-varying characteristics. Among them, the normalized energy distribution characteristics Represented as: ; In the formula, The normalized energy characteristic is represented by the underline, which indicates the normalization operation and reflects the proportion of the energy of this mode to the total energy. Indicates the first Energy of each mode; This represents the sum of the energies of all modes; The normalized bandwidth feature Represented as: ; In the formula, This represents the normalized bandwidth characteristic. Indicates the first Bandwidth of each mode; This represents the maximum bandwidth value across all modes; The normalized energy time-varying characteristics Represented as: ; In the formula, This represents the normalized energy change rate characteristic. Indicates the first The rate of energy change of each mode; This represents the maximum rate of energy change across all modes; Calculate the frequency weight and energy weight separately; The frequency weight is expressed as: ; In the formula, This indicates the frequency weight, reflecting the impact of frequency on noise requirements; Indicates the first Average frequency parameters of each mode; Indicates the target frequency center, controlling the central location of frequency preference; It represents the standard deviation of frequency weights and controls the width of frequency preference. Represents the natural exponential function; The energy weight is expressed as: ; In the formula, This represents the energy weight, reflecting the impact of energy on noise demand; This represents the normalized energy characteristic; The adaptive noise level is calculated based on the normalized energy distribution characteristics, the normalized bandwidth characteristics, and the normalized energy time-varying characteristics. The adaptive noise level is expressed as follows: ; In the formula, Indicates the first The original adaptive noise level of each modality, superscript Represents the initial calculated value without constraints; It represents the baseline noise level and controls the overall noise intensity; it is usually set to the initial noise standard deviation. This represents the energy suppression factor, reflecting the suppressive effect of energy on noise; This represents the bandwidth enhancement factor, reflecting the enhancement effect of bandwidth on noise. This represents the rate of change enhancement factor, reflecting the enhancing effect of the energy change rate on noise; This represents the frequency weight, which is used in the calculation as a frequency modulation factor. Establish noise level constraints; that is, ; In the formula, This represents the adaptive noise level after constraints. Indicates the original adaptive noise level; Indicates the minimum noise level constraint, subscript This represents the minimum value to prevent insufficient decomposition due to excessively low noise. Indicates the maximum noise level constraint, subscript This indicates the maximum value to prevent excessive noise from causing excessive aliasing. and These represent operations to retrieve the minimum and maximum values, respectively.

[0012] In some embodiments, the extraction module is further configured to establish a mapping relationship between noise and CEEMDAN decomposition levels based on the adaptive noise level; that is... ; In the formula, Indicates the first The noise level used in layered Ceemdan decomposition. This represents the IMF component index of CEEMDAN, with a value range of [value range missing]. ; The function ensures that the index does not go out of range when When using ; Indicates the final total number of IMF components; Adaptive noise level calculated using VMD characteristics A noise-assisted signal is constructed using the initial noise level instead of the fixed parameters used in traditional CEEMDAN. ,Right now ; In the formula, Indicates the first The signal after adding noise in the next implementation; Represents the original signal; This indicates the noise level used in the first layer of decomposition; Indicates the first The white noise sequence of this experiment; in superscript. This indicates that CEEMDAN implements indexing, and the value range is... , used to indicate the r-th implementation; This represents the total number of times CEEMDAN was implemented; For each noise-assisted signal, apply EMD to extract the first IMF: ; In the formula, Indicates the first The first IMF component obtained in this implementation; This indicates the extraction of the EMD operator for the first IMF; The final first IMF is obtained by ensemble averaging: ; In the formula, This represents the ensemble average of the first IMF component; the overline indicates the averaging operation. Calculate the residuals after the first decomposition: ; In the formula, This represents the residual signal after the first decomposition; Constructing noise-assisted signals ,Right now ; In the formula, Indicates the first The first implementation The signal after adding white noise to the layer residual; Indicates the first The residual signal after the decomposition; Indicates the first The noise level used in layer decomposition; Represents white noise The first result obtained by performing EMD operation One mode; Iterative extraction of IMF components, i.e. ; In the formula, Indicates the first The first implementation obtained the th One IMF component; Calculate the set average, i.e. ; In the formula, Indicates the first The ensemble average of the IMF components; Update residuals ,Right now ; In the formula, Indicates the first The residual signal after the decomposition.

[0013] In some embodiments, the establishing module is further configured to establish a residual amplitude stopping condition, i.e. In the formula, Residual The Hilbert transform amplitude; Indicates the stopping threshold; Establish the stopping condition for the extreme points of the residuals, i.e. ; In the formula, The first difference representing the residual is defined as follows: ; Represents a symbolic function; This represents the number of minimum extreme points; the iteration terminates when the residual amplitude stopping condition or the residual extreme point stopping condition is met, and the iteration number at termination is recorded as . , This is the final residual.

[0014] In some embodiments, the reconstructed signal is represented as: ; In the formula, Represents the raw IMU signal; This represents the sum of all IMF components; This represents the final residual.

[0015] This invention provides a VMD-CEEMDAN-driven IMU adaptive noise reduction device, comprising: a memory for storing executable instructions; and a processor for implementing the VMD-CEEMDAN-driven IMU adaptive noise reduction method by executing the executable instructions stored in the memory.

[0016] This invention provides a computer-readable storage medium storing executable instructions, which, when executed by a processor, implement the aforementioned VMD-CEEMDAN-driven IMU adaptive noise reduction method.

[0017] Compared with the prior art, the technical solution of the present invention has the following advantages: (1) To address the problem of mode aliasing caused by the fixed noise parameters in the traditional CEEMDAN algorithm, this invention combines the frequency domain analysis of variational mode decomposition with the complete set empirical mode decomposition, and establishes an adaptive noise parameter optimization mechanism based on the inherent frequency domain structure of the signal. By extracting the energy distribution, frequency characteristics, and bandwidth features of each mode, the optimal noise level is dynamically determined for each decomposition level, fundamentally solving the parameter setting problem, effectively avoiding mode aliasing, and significantly improving the signal decomposition quality and noise reduction effect.

[0018] (2) To address the problems of high computational complexity and poor real-time performance of existing algorithms, this invention significantly reduces the number of iterations and convergence time of the complete set empirical mode decomposition algorithm through an adaptive strategy guided by variational mode decomposition pre-analysis, improving computational efficiency by more than 50% compared to traditional methods. This method can meet the dual requirements of high-frequency data processing and real-time decision-making for UAV swarms.

[0019] (3) To address the lack of scientific rigor in setting termination conditions in traditional methods, this invention establishes a dual termination condition judgment mechanism based on residual amplitude analysis and signal extreme point statistics using Hilbert transform. This mechanism can adaptively determine the optimal termination time according to the actual decomposition state of the signal, effectively avoiding over-decomposition or insufficient decomposition, and ensuring the accuracy of signal reconstruction and the stability of the algorithm. Attached Figure Description

[0020] Figure 1 This is a flowchart illustrating a VMD-CEEMDAN-driven IMU adaptive noise reduction method provided by the present invention. Figure 2 This is a schematic diagram of the VMD-assisted CEEMDAN algorithm provided by the present invention; Figure 3 This is a diagram of the VMD-assisted CEEMDAN signal decomposition system architecture provided by the present invention. Figure 4 This is a diagram showing the IMF component decomposition results provided by the present invention; Figure 5 A schematic diagram of the composition structure of the VMD-CEEMDAN-driven IMU adaptive noise reduction device provided by the present invention; Figure 6 A schematic diagram of the composition structure of the electronic device provided by the present invention; Figure label: Figure 2 The module consists of: 1-Signal preprocessing and normalization module; 2-VMD decomposition and frequency domain characteristic extraction module; 3-Multidimensional feature extraction and normalization module; 4-Adaptive noise parameter calculation module; 5-IMF iterative extraction module; 6-Stop condition judgment module; 7-Signal reconstruction module; 8-Output IMF component set module. Figure 3 In the diagram: 9-Processor; 10-Memory; 11-Functional module group; 12-VMD decomposition module; 13-Feature extraction module; 14-CEEMDAN decomposition module; 15-Adaptive parameter calculation module; 16-Parallel computing unit; 17-Signal processing system. Detailed Implementation

[0021] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings. The described embodiments should not be regarded as limitations on the present invention. All other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0022] In the following description, references to "some embodiments" refer to a subset of all possible embodiments; however, it is understood that "some embodiments" may be the same or different subsets of all possible embodiments and may be combined with each other without conflict. Unless otherwise defined, all technical and scientific terms used in the embodiments of the invention have the same meaning as commonly understood by one of ordinary skill in the art to which the embodiments of the invention pertain. The terminology used in the embodiments of the invention is for the purpose of describing the embodiments of the invention only and is not intended to limit the invention.

[0023] The following describes an exemplary application of the VMD-CEEMDAN-driven IMU adaptive noise reduction device according to embodiments of the present invention. The VMD-CEEMDAN-driven IMU adaptive noise reduction device provided in these embodiments can be implemented as a terminal or a server. In one implementation, the VMD-CEEMDAN-driven IMU adaptive noise reduction device can be implemented as various types of terminals such as laptops, tablets, desktop computers, and mobile devices. In another implementation, the VMD-CEEMDAN-driven IMU adaptive noise reduction device can also be implemented as a server. The server can be an independent physical server, a server cluster or distributed system composed of multiple physical servers, or a cloud server providing basic cloud computing services such as cloud services, cloud databases, cloud computing, cloud functions, cloud storage, network services, cloud communication, middleware services, domain name services, security services, content delivery networks (CDNs), and big data and artificial intelligence platforms. The terminal and server can be directly or indirectly connected via wired or wireless communication, which is not limited in these embodiments. The following describes an exemplary application of the VMD-CEEMDAN-driven IMU adaptive noise reduction device when implemented as a server.

[0024] This invention provides a VMD-CEEMDAN-driven IMU adaptive noise reduction method, see [link to documentation]. Figure 1 , Figure 1 This is a flowchart illustrating a VMD-CEEMDAN-driven IMU adaptive noise reduction method provided in an embodiment of the present invention, which will be combined with... Figure 1 The steps shown are explained.

[0025] Step S110: The raw IMU signals acquired by the IMU sensor are sequentially standardized to obtain standardized IMU signals.

[0026] In some embodiments, the raw IMU signal, or Inertial Measurement Unit, is a sensor component capable of measuring the triaxial acceleration and triaxial angular velocity of an object. The raw IMU signal is the raw data signal reflecting the motion state of the object acquired by the sensor, typically including acceleration and angular velocity signals.

[0027] In some embodiments, standardization refers to preprocessing the raw IMU signal to convert it into a standard form with specific statistical characteristics. This is generally achieved by removing the signal's mean and normalizing its variance to a specific value (such as 1). The purpose is to eliminate the influence of differences in dimensions and amplitude ranges between different signals, providing a unified benchmark for subsequent signal decomposition and other processing, thereby improving the accuracy and stability of the processing.

[0028] Step S120: The variational mode decomposition algorithm is used to adaptively decompose the standardized IMU signal in the frequency domain to obtain the variational mode decomposition result.

[0029] In some embodiments, the Variational Mode Decomposition (VMD) algorithm is an adaptive signal decomposition method that decomposes the original signal into several modal components (i.e., variational mode decomposition results) with specific frequency domain characteristics by constructing and solving a variational problem. These modal components have characteristics such as finite bandwidth and center frequency.

[0030] Step S130: Extract the energy distribution characteristics, average frequency parameters, bandwidth characteristics, and energy time-varying characteristics of each mode from the variational mode decomposition results.

[0031] In some embodiments, energy distribution characteristics refer to the distribution of energy of each modal component in the time or frequency dimension, such as the intensity variation of energy in different time periods, the degree of concentration in different frequency ranges, etc.

[0032] In some embodiments, the average frequency parameter is a parameter obtained by statistically analyzing the frequency characteristics of each modal component. It refers to the average frequency value of the modal component within its existence time or frequency range, which can roughly reflect the main frequency position of the modal component and is one of the key parameters for describing the frequency characteristics of the modal component.

[0033] In some embodiments, the bandwidth feature represents the width of the frequency range occupied by each modal component, that is, the difference between the highest and lowest frequencies of the modal component. This feature reflects the size of the frequency distribution range of the modal components and is an important indicator for measuring the degree of frequency dispersion of the modal components.

[0034] In some embodiments, energy time-varying characteristics refer to the characteristics of how the energy of each modal component changes over time, such as the trend of energy increase or decrease over time, fluctuation frequency, etc.; it can reflect the changes in the energy state of modal components at different times, which is of great significance for analyzing the dynamic characteristics of signals.

[0035] Step S140: Based on the energy distribution characteristics, the average frequency parameter, the bandwidth characteristics, and the time-varying energy characteristics, construct an accurate mapping relationship from the inherent frequency domain structure of the signal to the noise intensity parameter; and by considering the modal energy weights, frequency position distribution, and bandwidth characteristics, dynamically determine the adaptive noise level for each decomposition level of the adaptive noise complete set empirical mode decomposition.

[0036] In some embodiments, the adaptive noise level refers to a noise intensity parameter dynamically determined during the CEEMDAN decomposition process based on the signal characteristics (such as the energy distribution characteristics and average frequency parameters mentioned above). Different decomposition levels correspond to different adaptive noise levels, the purpose of which is to make the decomposition process more adaptable to the intrinsic characteristics of the signal and improve the accuracy of the decomposition results.

[0037] Step S150: Using the adaptive noise level, the intrinsic mode function components are extracted layer by layer by constructing multiple sets of noise-assisted signals and performing ensemble averaging.

[0038] In some embodiments, the noise-assisted signal refers to the signal obtained by adding the original signal to a noise signal of a specific intensity (the noise intensity is determined by an adaptive noise level) in CEEMDAN decomposition. Constructing multiple sets of noise-assisted signals and performing ensemble averaging can effectively suppress the influence of noise on the decomposition results and improve the stability and reliability of the intrinsic mode function components.

[0039] In some embodiments, ensemble averaging is a process of averaging the results obtained from the decomposition of multiple sets of noise-assisted signals. By ensemble averaging, the influence of random noise in different noise-assisted signals can be offset, making the obtained intrinsic mode function components closer to the true signal components.

[0040] In some embodiments, the intrinsic mode function component is a signal component with specific characteristics obtained from CEEMDAN decomposition, satisfying the following two conditions: over the entire signal length, the number of extrema and the number of zero-crossings are equal or differ by at most one; at any given time, the average value of the upper envelope formed by local maxima and the lower envelope formed by local minima is zero. The intrinsic mode function component can reflect the local characteristics of the signal at different time scales.

[0041] Step S160: Establish a comprehensive judgment mechanism based on residual signal Hilbert transform amplitude analysis and extreme point statistics; when the residual signal meets the preset amplitude threshold condition or extreme point number threshold condition, the decomposition process is automatically terminated.

[0042] In some embodiments, the residual signal is the signal component remaining after CEEMDAN decomposition. It represents the low-frequency trend portion of the original signal that cannot be described by the intrinsic mode function components. The characteristics of the residual signal can reflect whether the decomposition process is sufficient and are one of the important criteria for determining whether the decomposition has terminated.

[0043] In some embodiments, the Hilbert transform is a mathematical transformation that yields an analytic signal from the residual signal. The amplitude of the analytic signal (i.e., the Hilbert transform amplitude) reflects the instantaneous amplitude characteristics of the residual signal. By analyzing this amplitude, the energy strength and variation of the residual signal can be determined.

[0044] In some embodiments, extreme point statistics is a process of statistically analyzing the number and distribution of maxima and minima in the residual signal. The number and distribution of extreme points can reflect the fluctuation characteristics and complexity of the residual signal, and are an important basis for determining whether the residual signal still contains valuable information.

[0045] In some embodiments, the preset amplitude threshold condition refers to an amplitude threshold set in advance according to actual application requirements and signal characteristics. When the Hilbert transform amplitude of the residual signal meets this condition (such as being less than or equal to the threshold), it indicates that the energy of the residual signal is already very weak, and further decomposition may not yield valuable information.

[0046] In some embodiments, the extreme point number threshold condition refers to a pre-set extreme point number threshold. When the number of extreme points in the residual signal meets this condition (e.g., less than or equal to the threshold), it indicates that the fluctuation of the residual signal has become very smooth, and further decomposition is not very meaningful.

[0047] Step S170: Linearly reconstruct the intrinsic mode function components and the final residual to generate the reconstructed IMU signal.

[0048] In some embodiments, linear reconstruction refers to recombining all intrinsic mode function components obtained from decomposition with the final residual signal in a certain linear combination manner (usually by direct addition) to obtain the reconstructed IMU signal. This process can remove noise interference from the original signal and restore the true characteristics of the signal.

[0049] First, addressing the issue of mode aliasing caused by fixed noise parameters in the traditional CEEMDAN algorithm, this invention combines frequency domain analysis of variational mode decomposition with complete ensemble empirical mode decomposition to establish an adaptive noise parameter optimization mechanism based on the inherent frequency domain structure of the signal. By extracting the energy distribution, frequency characteristics, and bandwidth features of each mode, the optimal noise level is dynamically determined for each decomposition level, fundamentally solving the parameter setting problem, effectively avoiding mode aliasing, and significantly improving signal decomposition quality and noise reduction effect.

[0050] Secondly, addressing the issues of high computational complexity and poor real-time performance in existing algorithms, this invention significantly reduces the number of iterations and convergence time of the complete set empirical mode decomposition algorithm through an adaptive strategy guided by variational mode decomposition pre-analysis, improving computational efficiency by more than 50% compared to traditional methods. This method can meet the dual requirements of high-frequency data processing and real-time decision-making for UAV swarms.

[0051] Finally, addressing the lack of scientific rigor in setting termination conditions for traditional methods, this invention establishes a dual stopping condition judgment mechanism based on residual amplitude analysis and signal extreme point statistics using Hilbert transform. This mechanism adaptively determines the optimal termination time according to the actual decomposition state of the signal, effectively avoiding over-decomposition or insufficient decomposition, and ensuring the accuracy of signal reconstruction and the stability of the algorithm.

[0052] In some embodiments, step S110 above can be implemented by the following: According to the formula For the original IMU signal The standardized IMU signal is obtained by performing standardization processing. ; In the formula, It is the standard deviation of the signal, defined as , This indicates the signal length, i.e., the number of sampling points; This represents the signal sampling point index, with a value ranging from 1 to... ; Indicates the first The signal value at each sampling point; It is the signal mean. .

[0053] In some embodiments, step S130 can be implemented by steps S131 to S132: Step S131: Apply the VMD decomposition algorithm to the standardized IMU signal and solve the following variational problem to obtain the VMD decomposition result.

[0054] Step S132: Extract the energy distribution characteristics, the average frequency parameter, the bandwidth characteristics, and the energy time-varying characteristics from the VMD decomposition results.

[0055] The variational problem is: ; In the formula, The first value obtained from VMD decomposition is represented as... One modal component, This is the index of the modal component, with a value range of 1. , This indicates the number of VMD modes, representing how many modal components the signal is decomposed into; Indicates the first The center frequency of each mode, superscript The center indicates the central position of the mode in the frequency domain; This represents the Dirac impulse function, used to construct analytic signals; This represents the time partial derivative operator, used to calculate the time derivative of a signal; Indicates the convolution operation; Represents the imaginary unit; express The square of the norm; t represents the time variable, corresponding to the sampling time in discrete implementation; The phrase "subject to" refers to the constraints of a variational problem.

[0056] The VMD decomposition result is as follows: ; In the formula, This represents the bandwidth constraint parameter, used to control the bandwidth of VMD decomposition; the larger the value, the smaller the bandwidth. This represents the noise tolerance parameter, which controls the VMD's sensitivity to noise. The larger the value, the less sensitive it is to noise. This indicates whether a DC component is included, with a value of 0 (not included) or 1 (included). represents the convergence tolerance, controls the VMD iteration termination condition, and represents the relative error threshold.

[0057] The energy distribution characteristics Represented as: ; In the formula, This represents the energy of the p-th mode, reflecting the importance of that mode in the signal; This represents the amplitude of the p-th mode at the j-th sampling point.

[0058] The average frequency parameter Represented as: ; In the formula, Indicates the first The average frequency of each mode is expressed in Hertz (Hz). Indicates the first The mode in the th ... The instantaneous frequency at each sampling point; Used to convert angular frequency to Hertz frequency.

[0059] The bandwidth characteristics Represented as: ; In the formula, Indicates the first The bandwidth of each mode reflects the width of the frequency distribution; Indicates the first The mode in the th ... The instantaneous frequency at each sampling point; This represents the mean of instantaneous frequencies. The overline indicates the averaging operation, defined as: .

[0060] The energy time-varying characteristics Represented as: ; In the formula, Indicates the first The energy change rate of each mode reflects the degree of energy change in the time domain; This represents the size of the moving average window, typically taken as... ; Indicates the first Each modality in Energy within a local window centered on the superscript To represent a local area, it is defined as: , This represents the index of the sampling point within the window, with a value range of [value range missing]. arrive ; Indicates the first The mode in the th ... Amplitude at each sampling point; local energy mean Defined as: ; Indicates the first The average energy of all local windows for each mode; This represents the total number of windows.

[0061] In some embodiments, step S140 can be implemented by steps S141 to S144: Step S141: Normalize the energy distribution characteristics, the bandwidth characteristics, and the time-varying energy characteristics to obtain normalized energy distribution characteristics, normalized bandwidth characteristics, and normalized time-varying energy characteristics.

[0062] The normalized energy distribution characteristics Represented as: ; In the formula, The normalized energy characteristic is represented by the underline, which indicates the normalization operation and reflects the proportion of the energy of this mode to the total energy. Indicates the first Energy of each mode; It represents the sum of the energies of all modes.

[0063] The normalized bandwidth feature Represented as: ; In the formula, This represents the normalized bandwidth characteristic. Indicates the first Bandwidth of each mode; This represents the maximum bandwidth value across all modes.

[0064] The normalized energy time-varying characteristics Represented as: ; In the formula, This represents the normalized energy change rate characteristic. Indicates the first The rate of energy change of each mode; This represents the maximum rate of energy change across all modes.

[0065] Step S142: Calculate the frequency weight and energy weight respectively.

[0066] The frequency weight is expressed as: ; In the formula, This indicates the frequency weight, reflecting the impact of frequency on noise requirements; Indicates the first Average frequency parameters of each mode; Indicates the target frequency center, controlling the central location of frequency preference; It represents the standard deviation of frequency weights and controls the width of frequency preference. This represents the natural exponential function.

[0067] The energy weight is expressed as: ; In the formula, This represents the energy weight, reflecting the impact of energy on noise demand; This represents the normalized energy characteristics.

[0068] Step S143: Calculate the adaptive noise level based on the normalized energy distribution characteristics, the normalized bandwidth characteristics, and the normalized time-varying energy characteristics.

[0069] The adaptive noise level is expressed as follows: ; In the formula, Indicates the first The original adaptive noise level of each modality, superscript Represents the initial calculated value without constraints; It represents the baseline noise level and controls the overall noise intensity; it is usually set to the initial noise standard deviation. This represents the energy suppression factor, reflecting the suppressive effect of energy on noise; This represents the bandwidth enhancement factor, reflecting the enhancement effect of bandwidth on noise. This represents the rate of change enhancement factor, reflecting the enhancing effect of the energy change rate on noise; This represents the frequency weight, which is used in the calculation as a frequency modulation factor.

[0070] Step S144: Establish noise level constraints; that is, .

[0071] In the formula, This represents the adaptive noise level after constraints. Indicates the original adaptive noise level; Indicates the minimum noise level constraint, subscript This represents the minimum value to prevent insufficient decomposition due to excessively low noise. Indicates the maximum noise level constraint, subscript This indicates the maximum value to prevent excessive noise from causing excessive aliasing. and These represent operations to retrieve the minimum and maximum values, respectively.

[0072] In some embodiments, step S150 can be implemented by steps S151 to S159: Step S151: Based on the adaptive noise level, establish a mapping relationship between noise and CEEMDAN decomposition levels; that is, ; In the formula, Indicates the first The noise level used in layered Ceemdan decomposition. This represents the IMF component index of CEEMDAN, with a value range of [value range missing]. ; The function ensures that the index does not go out of range when When using ; This represents the final total number of IMF components.

[0073] Step S152, using the adaptive noise level calculated based on VMD characteristics. A noise-assisted signal is constructed using the initial noise level instead of the fixed parameters used in traditional CEEMDAN. ,Right now ; In the formula, Indicates the first The signal after adding noise in the next implementation; Represents the original signal; This indicates the noise level used in the first layer of decomposition; Indicates the first The white noise sequence of this experiment; in superscript. This indicates that CEEMDAN implements indexing, and the value range is... , used to indicate the r-th implementation; This represents the total number of times CEEMDAN has been implemented.

[0074] Step S153: Apply EMD to each noise-assisted signal to extract the first IMF: ; In the formula, Indicates the first The first IMF component obtained in this implementation; This indicates the extraction of the EMD operator for the first IMF.

[0075] Step S154: Obtain the final first IMF through ensemble averaging. ; In the formula, This represents the ensemble average of the first IMF component; the overline indicates the averaging operation.

[0076] Step S155, calculate the residuals after the first decomposition: ; In the formula, This represents the residual signal after the first decomposition.

[0077] Step S156: Construct a noise-assisted signal ,Right now ; In the formula, Indicates the first The first implementation The signal after adding white noise to the layer residual; Indicates the first The residual signal after the decomposition; Indicates the first The noise level used in layer decomposition; Represents white noise The first result obtained by performing EMD operation One modality.

[0078] Step S157, iteratively extract the IMF components, i.e. ; In the formula, Indicates the first The first implementation obtained the th One IMF component.

[0079] Step S158, calculate the set average, i.e. ; In the formula, Indicates the first The ensemble average of the IMF components.

[0080] Step S159, update residuals ,Right now ; In the formula, Indicates the first The residual signal after the decomposition.

[0081] In some embodiments, step S160 can be implemented by steps S161 to S162: Step S161, establish the residual amplitude stopping condition, i.e. ; In the formula, Residual The Hilbert transform amplitude; This indicates the stop threshold.

[0082] Step S162, establish the stopping condition for the extreme point of the residual, that is ; In the formula, The first difference representing the residual is defined as follows: ; Represents a symbolic function; This represents the number of minimum extreme points; the iteration terminates when the residual amplitude stopping condition or the residual extreme point stopping condition is met, and the iteration number at termination is recorded as . , This is the final residual.

[0083] In some embodiments, the reconstructed signal is represented as: ; In the formula, Represents the raw IMU signal; This represents the sum of all IMF components; This represents the final residual.

[0084] The following will describe an exemplary application of the embodiments of the present invention in a practical application scenario.

[0085] The purpose of this invention is to provide an intelligent noise reduction processing method for IMU signals for UAV swarm navigation, comprising the following key steps: First, the raw signals acquired by the IMU sensor are preprocessed and standardized. Normalization is achieved by calculating the signal standard deviation, eliminating the influence of different amplitude ranges on subsequent analysis and providing a stable and reliable input basis for variational mode decomposition.

[0086] Second, a variational mode decomposition algorithm is used to adaptively decompose the preprocessed signal in the frequency domain. This algorithm decomposes the complex IMU signal into several eigenmode functions with clear physical meaning by solving a bandwidth-constrained variational optimization problem. Each mode has a compact support interval and a definite center frequency in the frequency domain.

[0087] Third, a multi-dimensional frequency domain feature extraction mechanism is constructed. The energy distribution characteristics, average frequency parameters, bandwidth characteristics, and time-varying energy properties of each mode are systematically extracted from the variational mode decomposition results. These feature parameters comprehensively characterize the signal's distribution and dynamic changes in the frequency domain, providing a sufficient information foundation for subsequent adaptive parameter calculation.

[0088] Fourth, an adaptive noise parameter calculation model is established. Based on the extracted multidimensional frequency domain features, a precise mapping relationship is constructed from the inherent frequency domain structure of the signal to the noise intensity parameters. By comprehensively considering modal energy weights, frequency position distribution, and bandwidth characteristics, the corresponding noise intensity parameters are dynamically determined for each decomposition level of the complete set of empirical mode decomposition, ensuring that each decomposition level can obtain an optimal noise environment that matches its frequency characteristics.

[0089] Fifth, an improved complete ensemble empirical mode decomposition based on adaptive parameters is performed. Using the calculated adaptive noise parameters, multiple sets of noise-assisted signals are constructed and ensemble averaging is performed to extract the intrinsic mode function components layer by layer. This process effectively suppresses mode aliasing caused by inappropriate noise parameters in traditional methods.

[0090] Sixth, implement dual stop condition judgment and decomposition control. Establish a comprehensive judgment mechanism based on residual signal Hilbert transform amplitude analysis and extreme point statistics. When the residual signal meets the preset amplitude threshold condition or extreme point number condition, the decomposition process is automatically terminated to ensure the sufficiency and accuracy of the decomposition.

[0091] Seventh, perform signal reconstruction and output processing. Linearly reconstruct all extracted intrinsic mode function components and the final residual to generate a high-quality, denoised IMU signal, providing reliable data support for subsequent navigation calculations and path prediction.

[0092] In terms of implementation, variational mode decomposition employs the alternating direction multiplier method to solve the constrained optimization problem, achieving adaptive signal decomposition through iterative updates of mode functions and center frequencies. Multidimensional feature extraction, through the organic combination of energy spectrum analysis, instantaneous frequency calculation, and bandwidth estimation, provides a comprehensive description of the signal's frequency domain characteristics. Adaptive noise parameter calculation establishes a comprehensive evaluation model considering frequency weights, energy weights, and rate of change weights, and sets reasonable parameter constraint boundaries. A dual stopping condition provides a reliable criterion for judging the degree of decomposition completion by evaluating the residual state in parallel.

[0093] This invention proposes a signal decomposition method based on Variational Mode Decomposition (VMD) assisted Complete Ensemble Empirical Mode Decomposition (CEEMDAN). The VMD algorithm pre-analyzes the signal to obtain frequency domain characteristic information, and then guides the CEEMDAN algorithm in selecting adaptive noise parameters, achieving high-precision decomposition of complex non-stationary signals.

[0094] The overall technical framework of this invention comprises two core innovative components: VMD frequency domain characteristic guidance and adaptive noise parameter calculation. These two core innovations, combined with targeted processing strategies, form a complete signal processing framework to address key challenges faced by traditional methods.

[0095] The technical implementation process of this invention is as follows: The original signal is preprocessed and normalized; the normalized signal is decomposed using the VMD algorithm to obtain the mode function and center frequency; modal features, including energy distribution, bandwidth, and frequency characteristics, are extracted based on the VMD decomposition results; an adaptive noise level is calculated based on the extracted features; an improved CEEMDAN decomposition is performed using the calculated adaptive noise level; an improved stopping criterion is applied to determine the decomposition termination condition; the signal is reconstructed and the final decomposition result is output. The entire processing flow is as follows: Figure 2 As shown, it specifically includes the following:

[0096] 1. Raw signal preprocessing: For the input signal ,in, Represents a time variable, with a value range of 1000. , Let be the total duration of the signal, corresponding to the sampling time in a discrete implementation. If the sampling frequency is , ,but Values , This represents the total number of signal sampling points.

[0097] For input signal Standardization is performed to eliminate the impact of amplitude differences on subsequent analysis, resulting in a normalized signal. : ; in, It is the standard deviation of the signal, defined as: , This indicates the signal length, i.e., the number of sampling points; This represents the signal sampling point index, with a value ranging from 1 to... ; Indicates the first The signal value at each sampling point; It is the signal mean: .

[0098] The purpose of this step is to ensure that signals with different amplitude ranges receive consistent processing results, providing a stable input for subsequent VMD decomposition.

[0099] 2. VMD Decomposition and Feature Extraction: 2.1 VMD decomposition of signals: For normalized signals Using VMD decomposition, the VMD algorithm solves the following variational problem: ; in, The first value obtained from VMD decomposition is represented as... One modal component, This is the index of the modal component, with a value range of 1. ; Indicates the first The center frequency of each mode, superscript The center indicates the central position of the mode in the frequency domain; This indicates the number of VMD modes, representing how many modal components the signal is decomposed into; This represents the Dirac impulse function, used to construct analytic signals; This represents the time partial derivative operator, used to calculate the time derivative of a signal; Indicates the convolution operation; The imaginary unit is indicated by bold to distinguish it from the index symbol; express The square of the norm.

[0100] This variational problem aims to find a set of modal components. and the corresponding center frequency This minimizes the bandwidth of each mode while ensuring the sum of all modes equals the original signal. Solving this variational problem yields the VMD decomposition result:

[0101] ; in, This represents the bandwidth constraint parameter, which controls the bandwidth of VMD decomposition; the larger the value, the smaller the bandwidth. This represents the noise tolerance parameter, which controls the VMD's sensitivity to noise. The larger the value, the less sensitive it is to noise. This indicates whether a DC component is included, with a value of 0 (not included) or 1 (included). represents the convergence tolerance, controls the VMD iteration termination condition, and represents the relative error threshold.

[0102] 2.2 Feature extraction from VMD decomposition results: The following features were extracted from the VMD decomposition results: 2.2.1 Extracting Energy Features Used to characterize the The energy magnitude of a mode is defined as the average of the squares of the signal of that mode: ; in, Indicates the first The energy of each mode reflects the importance of that mode in the signal; Indicates the first The mode in the th ... The amplitude at each sampling point; This is the sampling point index, with a value range of 1 to... , This is the signal length.

[0103] 2.2.2 Extracting frequency features Used to indicate the first The average frequency of each mode is calculated using the Hilbert transform.

[0104] 2.2.2.1 Constructing analytic signals : .in, Indicates the first The analytic signal of a modality is a complex signal; The imaginary unit; Indicates the first Hilbert transform of each mode.

[0105] 2.2.2.2 Calculation of instantaneous frequency : .in, Indicates the first The instantaneous frequency of each mode, superscript Indicates instantaneous; This represents the phase angle of the analytic signal; This represents the time derivative operation.

[0106] 2.2.2.3 Calculate the average frequency: .in, Indicates the first The average frequency of each mode is expressed in Hertz (Hz). Indicates the first The mode in the th ... The instantaneous frequency at each sampling point; Used to convert angular frequency to Hertz frequency.

[0107] 2.2.3 Extracting bandwidth features The range of variation of the p-th modal frequency is represented by the standard deviation of the instantaneous frequency. ; in, Indicates the first The bandwidth of each mode reflects the width of the frequency distribution; Indicates the first The mode in the th ... The instantaneous frequency at each sampling point; This represents the mean of instantaneous frequencies. The overline indicates the averaging operation, defined as: .

[0108] 2.2.4 Extracting the rate of energy change This is used to reflect the non-stationarity of the mode. Specifically, the local energy is calculated using a moving average window, and then its standard deviation is calculated.

[0109] ; in, Indicates the first The energy change rate of each mode reflects the degree of energy change in the time domain; This represents the size of the moving average window, typically taken as... ; Indicates the first Each modality in Energy within a local window centered on the superscript To represent a local area, it is defined as: , This represents the index of the sampling point within the window, with a value range of [value range missing]. arrive ; Indicates the first The mode in the th ... Amplitude at each sampling point; local energy mean Defined as: ;in, Indicates the first The average energy of all local windows for each mode; This represents the total number of windows.

[0110] 3. Adaptive noise level calculation: 3.1 Feature Normalization: First, the extracted features are normalized to unify their value range to the [0,1] interval, which facilitates subsequent calculations.

[0111] 3.1.1. Normalizing the energy yields... : .in, The normalized energy characteristic is represented by the underline, which indicates the normalization operation and reflects the proportion of the energy of this mode to the total energy. Indicates the first Energy of each mode; It represents the sum of the energies of all modes.

[0112] 3.1.2. Bandwidth normalization yields... : .in, This represents the normalized bandwidth characteristic. Indicates the first Bandwidth of each mode; This represents the maximum bandwidth value across all modes.

[0113] 3.1.3. Normalizing the rate of energy change yields... : .in, This represents the normalized energy change rate characteristic. Indicates the first The rate of energy change of each mode; This represents the maximum rate of energy change across all modes.

[0114] 3.2. Based on normalized features, calculate the adaptive noise level for each mode: 3.2.1 Calculate the frequency weights : .in, This indicates the frequency weight, reflecting the impact of frequency on noise requirements; Indicates the first The average frequency of each mode; Indicates the target frequency center, controlling the central location of frequency preference; It represents the standard deviation of frequency weights and controls the width of frequency preference. This represents the natural exponential function.

[0115] 3.2.2 Calculate the energy weight : .in, This represents the energy weight, reflecting the impact of energy on noise demand; This represents the normalized energy characteristics. Higher energy modes require less noise assistance.

[0116] 3.2.3 Calculate the adaptive noise level: ; in, Indicates the first The original adaptive noise level of each modality, superscript Represents the initial calculated value without constraints; It represents the baseline noise level and controls the overall noise intensity; it is usually set to the initial noise standard deviation. This represents the energy suppression factor, reflecting the suppressive effect of energy on noise; This represents the bandwidth enhancement factor, reflecting the enhancement effect of bandwidth on noise. This represents the rate of change enhancement factor, reflecting the enhancing effect of the energy change rate on noise; This represents the frequency weight, which is used in the calculation as a frequency modulation factor to provide frequency-based modulation.

[0117] 3.2.4 Establish noise level constraints: ; in, This represents the adaptive noise level after constraints. Indicates the original adaptive noise level; Indicates the minimum noise level constraint, subscript This represents the minimum value to prevent insufficient decomposition due to excessively low noise. Indicates the maximum noise level constraint, subscript This indicates the maximum value to prevent excessive noise from causing excessive aliasing. and These represent operations to retrieve the minimum and maximum values, respectively.

[0118] 3.2.5 Special Modal Processing: .

[0119] Here, a minimum noise level is set for the first mode (usually a high-frequency component), and zero is set for the last mode (usually a residual or trend component) to protect high-frequency information and low-frequency trend information. This indicates the noise level of the first mode; Indicates the first Noise level for each mode.

[0120] 4. Establish an improved CEEMDAN decomposition algorithm for VMD, hereinafter referred to as the VMD-CEEMDAN algorithm.

[0121] The core innovation of this invention lies in using the frequency domain characteristics obtained from VMD decomposition to adaptively adjust the noise level parameters in CEEMDAN, thereby solving problems such as mode aliasing and unstable decomposition quality caused by improper selection of noise parameters in the traditional CEEMDAN algorithm.

[0122] 4.1 Adaptive noise levels for each mode calculated based on Section 2 Establish the mapping relationship between noise and CEEMDAN decomposition levels: ; in, Indicates the first The noise level used in layered Ceemdan decomposition. This represents the IMF component index of CEEMDAN, with a value range of [value range missing]. ; The function ensures that the index does not go out of range when When using ; This represents the total number of final IMF components. This mapping strategy ensures that each layer of CEEMDAN decomposition uses a noise level that matches its frequency characteristics, avoiding the limitations of using a single fixed noise parameter in traditional methods.

[0123] 4.2 Extracting the first IMF component: Extract the first IMF component using the following steps: 4.2.1 Adaptive noise level calculated using VMD characteristics A noise-assisted signal is constructed using the initial noise level instead of the fixed parameters used in traditional CEEMDAN. : ; in, Indicates the first The signal after adding noise in this implementation, superscript Indicates the first This is the first implementation; Represents the original signal; This indicates the noise level used in the first layer of decomposition; Indicates the first The white noise sequence of this experiment; This indicates that CEEMDAN implements indexing, and the value range is... ; This indicates the number of times CEEMDAN has been implemented.

[0124] 4.2.2. Apply EMD to each noise-assisted signal to extract the first IMF: ;in, Indicates the first The first IMF component obtained in this implementation; This indicates the extraction of the EMD operator for the first IMF.

[0125] 4.2.3 Obtain the final first IMF through ensemble averaging: .in, This represents the ensemble average of the first IMF component; the overline indicates the averaging operation.

[0126] 4.2.4 Calculate the residuals after the first decomposition: .in, This represents the residual signal after the first decomposition.

[0127] 4.3. Subsequent IMF component extraction: for The following steps are executed iteratively: 4.3.1 Constructing a noise-assisted signal : ; in, Indicates the first The first implementation The signal after adding white noise to the layer residual; Indicates the first The residual signal after the decomposition; Indicates the first The noise level used in the layer decomposition is calculated based on VMD characteristics, rather than a fixed value in the traditional CEEMDAN. Represents white noise The first result obtained by performing EMD operation One modality.

[0128] 4.3.2 Iterative extraction of IMF components: ;in, Indicates the first The first implementation obtained the th One IMF component.

[0129] 4.3.3 Calculate the set average: .in, Indicates the first The ensemble average of the IMF components.

[0130] 4.3.4. Update residuals : .in, Indicates the first The residual signal after the decomposition.

[0131] 4.4 Improved stopping conditions: 4.4.1 Establishing the residual amplitude stopping condition: ; in, Residual The Hilbert transform amplitude; This indicates the stop threshold.

[0132] 4.4.2 Establishing the stopping condition for residual extreme points: ; in, The superscript represents the first difference of the residual. The first derivative or difference is defined as follows: ; Represents a symbolic function; This represents the number of minimum extreme points.

[0133] The iteration terminates when any of the above conditions are met, and the iteration number at termination is recorded. , This is the final residual.

[0134] 4.5 Signal Reconstruction: Recombining all IMF components and the final residual yields the reconstructed signal: ; in, Represents the original signal; This represents the sum of all IMF components; This represents the final residual.

[0135] Through the above steps, this invention organically combines the frequency domain characteristics extracted by VMD with CEEMDAN decomposition, achieving adaptive optimization of noise parameters and demonstrating better decomposition quality and stability compared to traditional methods. Compared to traditional CEEMDAN, this method shows significant improvements in mode separation capability, boundary effect suppression, and computational efficiency, providing a reliable tool for the analysis of various signal types. Figure 2 As shown, this algorithm achieves high-quality signal decomposition through key steps such as signal preprocessing and normalization (1), VMD decomposition and frequency domain characteristic extraction (2), multidimensional feature extraction and normalization (3), adaptive noise parameter calculation (4), IMF iterative extraction (5), stopping condition judgment (6), signal reconstruction (7), and output IMF component set (8).

[0136] 5. System Architecture and Implementation: This invention establishes an engineering implementation system based on the VMD-assisted CEEMDAN algorithm. Through modular design and parallel computing architecture, it effectively transforms the algorithm from a theoretical method into a practical application system. The system architecture fully considers the hardware characteristics and engineering requirements of modern computing environments, providing a complete technical solution for the efficient processing of complex non-stationary signals.

[0137] 5.1 System Overall Architecture Design: like Figure 3 As shown, the VMD-assisted CEEMDAN signal processing system (17) adopts a hierarchical modular architecture design. The entire system consists of three main components: functional module group (11), processor (9), and memory (10). The system architecture follows the principle of standardized interface design. The modules coordinate their work through standardized data interfaces and control buses to ensure the scalability and maintainability of the system.

[0138] Among them, the functional module group (11) is located at the upper layer of the system architecture and includes four core processing units: VMD decomposition module (12), feature extraction module (13), CEEMDAN decomposition module (14), and adaptive parameter calculation module (15). These functional modules are organized according to the logical flow of the algorithm. Each module undertakes a specific signal processing task, and the modules pass processing results through a standardized data flow interface.

[0139] 5.2 Parallel computing architecture.

[0140] like Figure 3As shown, the parallel computing unit (16) is configured with multiple independent processing cores, each of which can process one noise-implemented EMD decomposition task at a time. The parallel architecture makes full use of the statistical independence of each noise implementation in the CEEMDAN algorithm, and distributes the R noise-assisted calculations to different processing cores for parallel execution through task decomposition and load balancing mechanisms.

[0141] For the extraction of the first IMF component, the parallel computing unit performs the parallel EMD decomposition operation described by the formula: ; ParallelEM This represents the operator for parallel EMD decomposition of R-order noise. This parallel strategy utilizes a multi-core processor architecture to achieve task allocation and load balancing, ensuring full utilization of the computing resources of each processing core.

[0142] The subsequent iterative extraction process of IMF components also adopts a parallel architecture to implement the formula. Parallel computing.

[0143] The introduction of parallel processing allows the theoretical computation time for processing signals of length L to be reduced to 1 / C of the original time, where C represents the number of available processor cores.

[0144] Compared to traditional serial processing, this parallel implementation strategy significantly improves the algorithm's execution efficiency by maintaining the statistical properties of the CEEMDAN algorithm while employing a multi-core parallel computing architecture. Each parallel process independently processes the EMD decomposition implemented with noise, avoiding data dependencies between processes and ensuring the effectiveness of parallel computing and the accuracy of the algorithm results.

[0145] The processor module (9) serves as the computational execution center of the system, responsible for scheduling and controlling the collaborative work of various functional modules. The processor executes algorithm instructions stored in the program storage area to achieve precise control over functional modules such as VMD decomposition, feature extraction, parameter calculation, and CEEMDAN decomposition.

[0146] The memory module (10) is responsible for storing the computer program running in the system and the data during the processing. A bidirectional data channel is established between the processor and the memory to achieve efficient transmission of control instructions and storage of processing results.

[0147] The following will describe an exemplary application of the embodiments of the present invention in another practical application scenario.

[0148] I. Verification of Static Signal Decomposition Effect (1) Experimental setup This embodiment uses an ADIS16505 six-axis inertial measurement unit as the test signal source to verify the application effect of the technical solution of the present invention in sensor signal processing. The measurement unit includes a three-axis gyroscope and a three-axis accelerometer, with a rated sampling frequency of 100Hz. The equipment is preheated and stabilized for 40 minutes to ensure a consistent measurement environment. Signal data is continuously recorded for 90 minutes under constant temperature and static conditions, and outlier sampling points are removed using a standard outlier detection algorithm. A 2-minute continuous sampling segment (approximately 12,000 data points) is extracted from the complete dataset for algorithm verification, focusing on the technical effectiveness of the present invention in solving the problem of fixing noise parameters in traditional CEEMDAN systems.

[0149] (2) Analysis of processing results and performance indicators To comprehensively evaluate the performance of the method of this invention, the following three comparison methods were selected: the traditional CEEMDAN uses a fixed noise standard deviation coefficient of 0.2, the ICEEMDAN introduces a preliminary adaptive mechanism but still uses a fixed initial noise standard deviation coefficient of 0.2, and the VMD-CEEMDAN of this invention adopts a novel adaptive noise parameter strategy based on VMD frequency domain characteristic analysis.

[0150] The decomposition results are shown in Table 1. The computation time reflects the algorithm's execution efficiency, the number of IMFs represents the final number of intrinsic mode functions, MDQI represents the mode decomposition quality index, the orthogonality index measures the degree of independence between the IMF components, and the error compensation rate reflects the degree of performance improvement relative to the original sensor signal after decomposition and error compensation.

[0151] Table 1 Comparison of performance indicators of various CEEMDAN methods

[0152] As shown in Table 1, the method of this invention outperforms the comparative methods in all key performance indicators. The computation time is reduced by 52.9% compared to the traditional CEEMDAN and by 65.4% compared to ICEEMDAN, indicating that the VMD pre-analysis-guided adaptive noise strategy significantly improves the decomposition convergence speed. The number of IMFs is reduced from 16 to 13, effectively avoiding the over-decomposition phenomenon common in traditional methods. This is due to the accurate identification of the number of intrinsic modes of the signal by VMD frequency domain analysis. The orthogonality index is improved by 30.4%, demonstrating a significant improvement in the separation between modal components and reducing mode aliasing. The MDQI index reaches 0.7301, an improvement of 0.38% compared to the traditional CEEMDAN and 1.08% compared to ICEEMDAN, indicating a stable improvement in decomposition quality. The improvement in the modal decomposition quality index verifies the effectiveness of the VMD-assisted strategy in maintaining decomposition stability and accuracy. Furthermore, the improved error compensation rate indicates that the method of the present invention can more accurately identify and separate random error components in sensor signals. The modal feature information obtained through VMD frequency domain analysis guides CEEMDAN to achieve more effective error modeling and compensation, directly improving the accuracy and reliability of signal processing.

[0153] When comparing the processing performance with a scale of 12,000 data points, the method of this invention exhibits significant comprehensive advantages. VMD pre-analysis not only provides accurate frequency domain feature information, but more importantly, it establishes an adaptive mapping mechanism from frequency domain characteristics to noise parameters, enabling the modal components of each frequency band to be decomposed under the most suitable noise environment. This refined parameter adjustment strategy avoids the limitations of traditional methods with fixed noise parameters, significantly improving computational efficiency while ensuring decomposition accuracy. The decomposition results are as follows: Figure 4 As shown.

[0154] Figure 4 This paper presents the decomposition results of the IMU angular velocity signal using the VMD-CEEMDAN collaborative method. The original signal (top) is decomposed into 12 IMF components and 1 residual component. The decomposition results conform to a decreasing energy distribution characteristic. IMF1-IMF6 mainly contain high-frequency noise components, with energy decreasing from 4.50% to 0.20%. IMF7-IMF12 exhibit mid-to-low frequency characteristics, with their energy proportion gradually decreasing. The residual component (bottom) contains a low-frequency trend in the signal, accounting for 92.18% of the total energy. The decomposition results demonstrate that this method can effectively separate signal components at different frequency scales, achieving separation of noise and valid information.

[0155] Figure 5 This is a schematic diagram of the composition structure of the VMD-CEEMDAN-driven IMU adaptive noise reduction device provided in an embodiment of the present invention, as shown below. Figure 5As shown, the VMD-CEEMDAN-driven IMU adaptive noise reduction device 500 includes: a processing module 501, used to sequentially standardize the raw IMU signals acquired by the IMU sensor to obtain standardized IMU signals; a decomposition module 502, used to perform adaptive frequency domain decomposition on the standardized IMU signals using a variational mode decomposition algorithm to obtain variational mode decomposition results; an extraction module 503, used to extract the energy distribution characteristics, average frequency parameters, bandwidth characteristics, and energy time-varying characteristics of each mode from the variational mode decomposition results; and a determination module 504, used to construct a noise reduction model from the inherent frequency domain structure of the signal to the noise reduction model based on the energy distribution characteristics, the average frequency parameters, the bandwidth characteristics, and the energy time-varying characteristics. The precise mapping relationship of sound intensity parameters; and by considering modal energy weights, frequency position distribution and bandwidth characteristics, dynamically determining the adaptive noise level for each decomposition level of the adaptive noise complete set empirical mode decomposition; the extraction module 503 is also used to use the adaptive noise level to extract the intrinsic mode function components layer by layer by constructing multiple sets of noise auxiliary signals and performing ensemble averaging; the establishment module 505 is used to establish a comprehensive judgment mechanism based on residual signal Hilbert transform amplitude analysis and extreme point statistics; when the residual signal meets the preset amplitude threshold condition or extreme point number threshold condition, the decomposition process is automatically terminated; the generation module 506 is used to linearly reconstruct the intrinsic mode function components and the final residual to generate the reconstructed IMU signal.

[0156] It should be noted that the description of the apparatus in this embodiment is similar to that of the method embodiment described above, and has similar beneficial effects, therefore it will not be repeated. For technical details not disclosed in this apparatus embodiment, please refer to the description of the method embodiment of this invention for understanding.

[0157] It should be noted that, in the embodiments of the present invention, if the above-described VMD-CEEMDAN-driven IMU adaptive noise reduction method is implemented as a software functional module and sold or used as an independent product, it can also be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the embodiments of the present invention, or the part that contributes to related technologies, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a terminal to execute all or part of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), magnetic disks, or optical disks. Thus, the embodiments of the present invention are not limited to any specific hardware and software combination.

[0158] Correspondingly, embodiments of the present invention provide an electronic device, Figure 6 This is a schematic diagram of the composition structure of the electronic device provided in the embodiments of the present invention, such as... Figure 6 As shown, the electronic device 600 includes at least a processor 601 and a computer-readable storage medium 602 configured to store executable instructions, wherein the processor 601 generally controls the overall operation of the electronic device 600. The computer-readable storage medium 602 is configured to store instructions and applications executable by the processor 601, and may also cache data to be processed or processed by various modules in the processor 601 and the electronic device 600, and may be implemented using flash memory or random access memory (RAM).

[0159] This invention provides a storage medium storing executable instructions. When these executable instructions are executed by a processor, they cause the processor to perform the method provided in this invention, for example... Figure 1 The method shown.

[0160] In some embodiments, the storage medium may be a computer-readable storage medium, such as a ferromagnetic random access memory (FRAM), a read-only memory (ROM), a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), flash memory, magnetic surface memory, optical disc, or a compact disk-read-only memory (CD-ROM); or it may be a device that includes one or any combination of the above-mentioned memories.

[0161] In some embodiments, executable instructions may take the form of a program, software, software module, script, or code, written in any form of programming language (including compiled or interpreted languages, or declarative or procedural languages), and may be deployed in any form, including as a standalone program or as a module, component, subroutine, or other unit suitable for use in a computing environment.

[0162] As an example, executable instructions may, but do not necessarily, correspond to files in a file system. They may be stored as part of a file containing other programs or data, for example, in one or more scripts within a Hyper Text Markup Language (HTML) document, in a single file dedicated to the program in question, or in multiple collaborating files (e.g., files storing one or more modules, subroutines, or code sections). As an example, executable instructions may be deployed to execute on a single electronic device, or on multiple electronic devices located in one location, or on multiple electronic devices distributed across multiple locations and interconnected via a communication network.

[0163] The above description is merely an embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and scope of the present invention are included within the scope of protection of the present invention.

[0164] It should be understood that the phrase "one embodiment" or "an embodiment" throughout the specification means that a specific feature, structure, or characteristic related to the embodiment is included in at least one embodiment of the invention. Therefore, "in one embodiment" or "in an embodiment" appearing throughout the specification does not necessarily refer to the same embodiment. Furthermore, these specific features, structures, or characteristics can be combined in any suitable manner in one or more embodiments. It should be understood that in the various embodiments of the invention, the sequence numbers of the above-described processes do not imply a sequential 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 the invention. The sequence numbers of the above-described embodiments of the invention are merely descriptive and do not represent the superiority or inferiority of the embodiments.

[0165] It should be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes that element. In the several embodiments provided by this invention, it should be understood that the disclosed devices and methods can be implemented in other ways. The device embodiments described above are merely illustrative; for example, the division of units is only a logical functional division, and in actual implementation, there may be other division methods, such as: multiple units or components may be combined, or integrated into another system, or some features may be ignored or not performed.

[0166] The above description is merely an embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A VMD-CEEMDAN-driven IMU adaptive noise reduction method, characterized in that, The method includes: The raw IMU signals acquired by the IMU sensor are sequentially standardized to obtain standardized IMU signals; The variational mode decomposition algorithm is used to adaptively decompose the standardized IMU signal in the frequency domain to obtain the variational mode decomposition result; The energy distribution characteristics, average frequency parameters, bandwidth characteristics, and energy time-varying properties of each mode are extracted from the variational mode decomposition results. Based on the energy distribution characteristics, the average frequency parameter, the bandwidth characteristics, and the time-varying energy properties, a precise mapping relationship from the inherent frequency domain structure of the signal to the noise intensity parameter is constructed; and by considering the modal energy weights, frequency position distribution, and bandwidth characteristics, the adaptive noise level is dynamically determined for each decomposition level of the adaptive noise complete set empirical mode decomposition. Using the aforementioned adaptive noise level, multiple sets of noise-assisted signals are constructed and ensemble averaging is performed to extract the intrinsic mode function components layer by layer. A comprehensive judgment mechanism based on residual signal Hilbert transform amplitude analysis and extreme point statistics is established; the decomposition process is automatically terminated when the residual signal meets the preset amplitude threshold condition or extreme point number threshold condition. The intrinsic mode function components and the final residual are linearly reconstructed to generate the reconstructed IMU signal.

2. The method according to claim 1, characterized in that, The raw IMU signals acquired by the IMU sensor are sequentially standardized to obtain standardized IMU signals, including: According to the formula For the original IMU signal The standardized IMU signal is obtained by performing standardization processing. ; In the formula, It is the standard deviation of the signal, defined as , This indicates the signal length, i.e., the number of sampling points; This represents the signal sampling point index, with a value ranging from 1 to... ; Indicates the first The signal value at each sampling point; It is the signal mean. .

3. The method according to claim 1, characterized in that, The extraction of energy distribution characteristics, average frequency parameters, bandwidth characteristics, and time-varying energy properties of each mode from the variational mode decomposition results includes: The VMD decomposition algorithm is applied to the standardized IMU signal, and the following variational problem is solved to obtain the VMD decomposition result; The energy distribution characteristics, the average frequency parameters, the bandwidth characteristics, and the energy time-varying characteristics are extracted from the VMD decomposition results. The variational problem is: ; In the formula, The first term obtained from VMD decomposition is represented by the first term. One modal component, This is the index of the modal component, with a value range of 1. ; This indicates the number of VMD modes, representing how many modal components the signal is decomposed into; Indicates the first The center frequency of each mode, superscript The center indicates the central position of the mode in the frequency domain; This represents the Dirac impulse function, used to construct analytic signals; This represents the time partial derivative operator, used to calculate the time derivative of a signal; Indicates the convolution operation; Represents the imaginary unit; express The square of the norm; t represents the time variable, corresponding to the sampling time in discrete implementation; The phrase "bound to" refers to the constraints of a variational problem. This represents the standardized IMU signal; The VMD decomposition result is as follows: ; In the formula, This represents the bandwidth constraint parameter, used to control the bandwidth of VMD decomposition; the larger the value, the smaller the bandwidth. This represents the noise tolerance parameter, which controls the VMD's sensitivity to noise. The larger the value, the less sensitive it is to noise. This indicates whether a DC component is included, with a value of 0 (not included) or 1 (included). This represents the convergence tolerance, controls the VMD iteration termination condition, and represents the relative error threshold. The energy distribution characteristics Represented as: ; In the formula, This represents the energy of the p-th mode, reflecting the importance of that mode in the signal; This indicates the signal length, i.e., the number of sampling points; This represents the amplitude of the p-th mode at the j-th sampling point; The average frequency parameter Represented as: ; In the formula, Indicates the first The average frequency of each mode is expressed in Hertz (Hz). Indicates the first The mode in the th ... The instantaneous frequency at each sampling point; Used to convert angular frequency to Hertz frequency; The bandwidth characteristics Represented as: ; In the formula, Indicates the first The bandwidth of each mode reflects the width of the frequency distribution; Indicates the first The mode in the th ... The instantaneous frequency at each sampling point; This represents the mean of instantaneous frequencies. The overline indicates the averaging operation, defined as: ; The time-varying characteristics of energy Represented as: ; In the formula, Indicates the first The energy change rate of each mode reflects the degree of energy change in the time domain; This represents the size of the moving average window, typically taken as... ; Indicates the first Each modality in Energy within a local window centered on the superscript To represent a local area, it is defined as: , This represents the index of the sampling point within the window, with a value range of [value missing]. arrive ; Indicates the first The mode in the th ... Amplitude at each sampling point; local energy mean Defined as: ; Indicates the first The average energy of all local windows for each mode; This represents the total number of windows.

4. The method according to claim 1, characterized in that, The process involves constructing a precise mapping relationship from the inherent frequency domain structure of the signal to the noise intensity parameter based on the energy distribution characteristics, the average frequency parameter, the bandwidth characteristics, and the time-varying energy characteristics; and dynamically determining the corresponding noise intensity parameters for each decomposition level of the adaptive noise complete set empirical mode decomposition by considering modal energy weights, frequency position distribution, and bandwidth characteristics, including: The energy distribution characteristics, the bandwidth characteristics, and the energy time-varying characteristics are normalized respectively to obtain normalized energy distribution characteristics, normalized bandwidth characteristics, and normalized energy time-varying characteristics. Among them, the normalized energy distribution characteristics Represented as: ; In the formula, The normalized energy characteristic is represented by the underline, which indicates the normalization operation and reflects the proportion of the energy of this mode to the total energy. Indicates the first Energy of each mode; This represents the sum of the energies of all modes; The normalized bandwidth feature Represented as: ; In the formula, This represents the normalized bandwidth characteristic. Indicates the first Bandwidth of each mode; This represents the maximum bandwidth value across all modes; The normalized energy time-varying characteristics Represented as: ; In the formula, This represents the normalized energy change rate characteristic. Indicates the first The rate of energy change of each mode; This represents the maximum rate of energy change across all modes; Calculate the frequency weight and energy weight separately; The frequency weight is expressed as: ; In the formula, This indicates the frequency weight, reflecting the impact of frequency on noise requirements; Indicates the first Average frequency parameters of each mode; Indicates the target frequency center, controlling the central location of frequency preference; It represents the standard deviation of frequency weights and controls the width of frequency preference. Represents the natural exponential function; The energy weight is expressed as: ; In the formula, This represents the energy weight, reflecting the impact of energy on noise demand; This represents the normalized energy characteristic. The adaptive noise level is calculated based on the normalized energy distribution characteristics, the normalized bandwidth characteristics, and the normalized energy time-varying characteristics. The adaptive noise level is expressed as follows: ; In the formula, Indicates the first The original adaptive noise level of each modality, superscript Represents the initial calculated value without constraints; It represents the baseline noise level and controls the overall noise intensity; it is usually set to the initial noise standard deviation. This represents the energy suppression factor, reflecting the suppressive effect of energy on noise; This represents the bandwidth enhancement factor, reflecting the enhancement effect of bandwidth on noise. This represents the rate of change enhancement factor, reflecting the enhancing effect of the energy change rate on noise; This represents the frequency weight, which is used in the calculation as a frequency modulation factor. Establish noise level constraints; that is, ; In the formula, This represents the adaptive noise level after constraints. Indicates the original adaptive noise level; This represents a minimum noise level constraint to prevent insufficient decomposition due to excessively low noise levels. This represents the maximum noise level constraint to prevent excessive noise from causing excessive aliasing.

5. The method according to claim 1, characterized in that, The step of utilizing the adaptive noise level to extract intrinsic mode function components layer by layer by constructing multiple sets of noise-assisted signals and performing ensemble averaging includes: Based on the adaptive noise level, a mapping relationship is established between noise and CEEMDAN decomposition levels; that is... ; In the formula, Indicates the first The noise level used in layered Ceemdan decomposition. This represents the IMF component index of CEEMDAN, with a value range of [value range missing]. ; The function ensures that the index does not go out of range when When using ; Indicates the final total number of IMF components; Adaptive noise level calculated using VMD characteristics A noise-assisted signal is constructed using the initial noise level instead of the fixed parameters used in traditional CEEMDAN. ,Right now ; In the formula, Indicates the first The signal after adding noise in the next implementation; Represents the raw IMU signal; This indicates the noise level used in the first layer of decomposition; Indicates the first The white noise sequence of this experiment; in superscript. This indicates that CEEMDAN implements indexing, and the value range is... , used to indicate the r-th implementation; This represents the total number of times CEEMDAN was implemented; For each noise-assisted signal, apply EMD to extract the first IMF: ; In the formula, Indicates the first The first IMF component obtained in this implementation; This indicates the extraction of the EMD operator for the first IMF; The final first IMF is obtained by ensemble averaging: ; In the formula, This represents the ensemble average of the first IMF component; the overline indicates the averaging operation. Calculate the residuals after the first decomposition: ; In the formula, This represents the residual signal after the first decomposition; Constructing noise-assisted signals ,Right now ; In the formula, Indicates the first The first implementation The signal after adding white noise to the layer residual; Indicates the first The residual signal after the decomposition; Indicates the first The noise level used in layer decomposition; Represents white noise The first result obtained by performing EMD operation One mode; Iterative extraction of IMF components, i.e. ; In the formula, Indicates the first The first implementation obtained the th One IMF component; Calculate the set average, i.e. ; In the formula, Indicates the first The ensemble average of the IMF components; Update residuals ,Right now ; In the formula, Indicates the first The residual signal after the decomposition.

6. The method according to claim 1, characterized in that, The establishment of a comprehensive judgment mechanism based on residual signal Hilbert transform amplitude analysis and extreme point statistics includes: Establish the residual amplitude stopping condition, i.e. ; In the formula, Residual The Hilbert transform amplitude; Indicates the stopping threshold; This indicates the signal length, i.e., the number of sampling points; This represents the signal sampling point index, with a value ranging from 1 to... ; Establish the stopping condition for the extreme points of the residuals, i.e. ; In the formula, The first difference representing the residual is defined as follows: ; Represents a symbolic function; This represents the number of minimum extreme points; the iteration terminates when the residual amplitude stopping condition or the residual extreme point stopping condition is met, and the iteration number at termination is recorded as . , This is the final residual.

7. The method according to claim 1, characterized in that, The reconstructed signal is represented as: ; In the formula, Represents the raw IMU signal; This represents the sum of all IMF components; This represents the final residual.

8. A VMD-CEEMDAN-driven IMU adaptive noise reduction device, characterized in that, The device includes: The processing module is used to perform standardization processing on the raw IMU signals acquired by the IMU sensor in sequence to obtain standardized IMU signals; The decomposition module is used to perform adaptive frequency domain decomposition on the standardized IMU signal using a variational mode decomposition algorithm to obtain the variational mode decomposition result. The extraction module is used to extract the energy distribution characteristics, average frequency parameters, bandwidth characteristics, and energy time-varying characteristics of each mode from the variational mode decomposition results. The determination module is used to construct an accurate mapping relationship from the inherent frequency domain structure of the signal to the noise intensity parameter based on the energy distribution characteristics, the average frequency parameter, the bandwidth characteristics, and the time-varying energy characteristics; and to dynamically determine the adaptive noise level for each decomposition level of the adaptive noise complete set empirical mode decomposition by considering modal energy weights, frequency position distribution, and bandwidth characteristics. The extraction module is also used to extract the intrinsic mode function components layer by layer by constructing multiple sets of noise-assisted signals and performing ensemble averaging processing using the adaptive noise level. A module is established to create a comprehensive judgment mechanism based on residual signal Hilbert transform amplitude analysis and extreme point statistics; the decomposition process is automatically terminated when the residual signal meets the preset amplitude threshold condition or extreme point number threshold condition. The generation module is used to linearly reconstruct the intrinsic mode function components and the final residual to generate the reconstructed IMU signal.

9. A VMD-CEEMDAN-driven IMU adaptive noise reduction device, characterized in that, include: Memory, used to store executable instructions; The processor, when executing executable instructions stored in the memory, implements the VMD-CEEMDAN-driven IMU adaptive noise reduction method according to any one of claims 1 to 7.

10. A computer-readable storage medium storing executable instructions for causing a processor to execute the executable instructions to implement the VMD-CEEMDAN driven IMU adaptive noise reduction method according to any one of claims 1 to 7.

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