With the rapid development of Artificial Intelligence of Things(AIoT)technologies,the security of Industrial Internet of Things(IIoT)data faces increasing challenges,particularly in time series anomaly detection.IIoT ...With the rapid development of Artificial Intelligence of Things(AIoT)technologies,the security of Industrial Internet of Things(IIoT)data faces increasing challenges,particularly in time series anomaly detection.IIoT data are typically scarce in abnormal samples and noisy,making unsupervised learning a common solution.The security challenges of IIoT data in AIoT environments require robust unsupervised anomaly detection methods.While Variational Autoencoders(VAEs)excel in noise resilience,they face two critical challenges in IIoT data:difficulties in single-variable time-series modeling and conflicts between static prior assumptions and dynamic temporal features.To address these challenges,we propose the Greater Cane Rat Algorithm-enhanced FourierWavelet Conditional Variational Autoencoder(GCRA-FWVAE).Our method introduces a time-frequency dualbranch architecture that synergistically combines wavelet transforms for localized transient feature extraction and Fourier transforms for global spectral characterization.These complementary representations jointly regulate the Conditional Variational Autoencoder(CVAE)reconstruction process,effectively preserving critical anomaly signatures while suppressing noise interference.The architecture is further optimized through bioinspired Greater Cane Rat Algorithm(GCRA)to improve adaptive learning capabilities.Extensive validation on the Yahoo benchmark indicates state-of-the-art performance,achieving an F1-score of 93.6%(an improvement of 4.5% over baseline VAEs)and a precision of 95.1%.These improvements significantly increase anomaly detection accuracy and robustness,particularly in the AIoT environment,where it effectively handles more complex and dynamic industrial data.展开更多
Arrhythmias are a frequently occurring phenomenon in clinical practice,but how to accurately dis-tinguish subtle rhythm abnormalities remains an ongoing difficulty faced by the entire research community when conductin...Arrhythmias are a frequently occurring phenomenon in clinical practice,but how to accurately dis-tinguish subtle rhythm abnormalities remains an ongoing difficulty faced by the entire research community when conducting ECG-based studies.From a review of existing studies,two main factors appear to contribute to this problem:the uneven distribution of arrhythmia classes and the limited expressiveness of features learned by current models.To overcome these limitations,this study proposes a dual-path multimodal framework,termed DM-EHC(Dual-Path Multimodal ECG Heartbeat Classifier),for ECG-based heartbeat classification.The proposed framework links 1D ECG temporal features with 2D time–frequency features.By setting up the dual paths described above,the model can process more dimensions of feature information.The MIT-BIH arrhythmia database was selected as the baseline dataset for the experiments.Experimental results show that the proposed method outperforms single modalities and performs better for certain specific types of arrhythmias.The model achieved mean precision,recall,and F1 score of 95.14%,92.26%,and 93.65%,respectively.These results indicate that the framework is robust and has potential value in automated arrhythmia classification.展开更多
Understanding how rock slopes respond to blasting loads is crucial for maintaining excavation safety and slope stability.Nevertheless,the spatiotemporal evolution,nonlinear dependence on blasting parameters,and predic...Understanding how rock slopes respond to blasting loads is crucial for maintaining excavation safety and slope stability.Nevertheless,the spatiotemporal evolution,nonlinear dependence on blasting parameters,and predictive behavior of dominant frequency responses in slope vibrations remain insufficiently understood and quantified.This study combines time-frequency analysis with machine learning to explore how the dominant frequency(fd)evolves in slopes under blasting.Continuous Wavelet Transform(CWT)was employed to characterize the temporal-frequency evolution of vibration signals,revealing that the dominant frequency exhibits strong spatial dependence and nonlinear variability influenced by blasting parameters and rock mass structures.Three machine learning models,namely Back Propagation Neural Network(BP),Support Vector Machine(SVM),and Random Forest(RF),were developed to predict fd based on 1,000 monitoring samples obtained from numerical and field simulations.Among them,the RF model achieved the highest prediction accuracy,with mean absolute percentage errors(MAPE)below 15%,demonstrating strong robustness and generalization capability.Our analysis shows that external excitation factors,especially the loading frequency(fd),mainly control the frequency response,while internal controlling factors,such as spatial position,lithological variation,and mechanical heterogeneity,modulate localized frequency amplification and energy redistribution.The results reveal that fd tends to decrease with elevation and distance from the blasting source,whereas structural planes and weathered zones induce high-frequency amplification due to scattering and modal coupling effects.This study offers a new framework combining time-frequency analysis and machine learning to measure the nonlinear interaction between blasting and rock mass response,offering new insights for dynamic stability evaluation and hazard mitigation in complex rock slope systems.展开更多
The state-of-the-art optical atomic clocks and the time-frequency signal transmission open a fresh field for gravity potential(geopotential)determination.Various methods,including optical fiber frequency transfer,sate...The state-of-the-art optical atomic clocks and the time-frequency signal transmission open a fresh field for gravity potential(geopotential)determination.Various methods,including optical fiber frequency transfer,satellite two-way,satellite common-view,satellite carrier phase,VLBI,tri-frequency combination,and dual-frequency combination,were developed to determine the geopotential differences using optical atomic clocks and then determine the geopotential at station B based on the geopotential at station A.This review elaborates the principles,methods,scientific objectives,applications,and relevant research trends of geopotential determination based on time-frequency signals.展开更多
In recent years,anomaly detection in Wireless Sensor Networks(WSNs)has been widely studied using Graph Neural Networks and Transformer-based methods.However,in multi-node and multi-modal data scenarios,these approache...In recent years,anomaly detection in Wireless Sensor Networks(WSNs)has been widely studied using Graph Neural Networks and Transformer-based methods.However,in multi-node and multi-modal data scenarios,these approaches still face challenges such as insufficient extraction of spatiotemporal correlation features,limited modeling capabilities when relying solely on either time-domain or frequency-domain information,and high computational overhead.To address these issues,this work aims to develop an anomaly detection model that balances detection performance with computational efficiency,enabling effective identification of complex anomaly patterns.Specifically,we propose a time–frequency feature extraction method with topological information enhancement,topology-enhanced multi-modal spatio-temporal anomaly detection(TE-MSTAD).Building upon the Receptance Weighted Key Value(RWKV)model with linear complexity,a cross-modal feature extraction module is introduced to strengthen the modeling of multi-modal correlations.Meanwhile,adaptive adjacency matrices are constructed by integrating time–frequency features and combining outputs from different Graph Neural Networks,thereby enhancing topological information.Furthermore,a dual-branch structure is designed to jointly model time-domain and frequency-domain features,improving the extraction of complex anomaly characteristics.Experiments on both publicly available datasets and real-world collected data demonstrate that the proposed method achieves F1-scores of 92.52%and 93.28%,respectively,outperforming existing methods in detection performance and generalization capability.展开更多
Fault features in mechanical systems often manifest as transient impulses,which can be effectively analyzed using time-frequency analysis(TFA)methods.Recently,a new TFA technique known as the time-reassigned multi-syn...Fault features in mechanical systems often manifest as transient impulses,which can be effectively analyzed using time-frequency analysis(TFA)methods.Recently,a new TFA technique known as the time-reassigned multi-synchrosqueezing transform(TMssT)was proposed to capture these transient impulses for fault diagnosis.However,the TMSST,which is based on the short-time Fourier transform(STFT),suffers from unclear high-frequency re-presentations owing to the fixed sliding window used in the STFT.To address this limitation,the current study combined TMSST with the S-transform and a local maximum method to enhance the time-frequency representation for improved signal analysis.Furthermore,an extractive reconstruction algorithm that binds the maximum value of the spectral envelope is proposed for spectral decomposition.To validate the proposed technique,a simulated noise-added signal and four experimental bearing defect datasets were used.The results demonstrate that the proposed technique can effectively and accurately extract fault features from bearing signals regardless of whether the bearings operate under constant or varying speed conditions.This study offers a novel and efficient approach for fault diagnosis in mechanical systems with complex dynamic behaviors.展开更多
With the development of Sixth-Generation(6G)mobile communication technologies,Low Earth Orbit(LEO)satellite communication systems have become extremely important in mobile communications owing to their large coverage,...With the development of Sixth-Generation(6G)mobile communication technologies,Low Earth Orbit(LEO)satellite communication systems have become extremely important in mobile communications owing to their large coverage,high efficiency,and low cost.However,the high dynamic LEO satellite channels cause serious time-frequency dual selective fading,significantly impairing the performance of conventional single time or frequency domain synchronization algorithms and limiting their applicability.To address these challenges,this paper proposes a synchronization algorithm based on Linear Frequency Modulation(LFM)signals and the Fractional Fourier Transform(FRFT).Exploiting the inherent robustness of LFM signals against frequency deviations and multipath effects,coupled with their energy concentration property in the optimal fractional Fourier domain,the proposed algorithm enables efficient synchronization with enhanced resilience to time-frequency variations.Furthermore,LFM preamble sequences are optimally designed for diverse channel conditions.This work presents a theoretical analysis of the time-frequency nonstationary characteristics of LEO satellite channels and discusses the performance limitations of traditional synchronization algorithms.The proposed integrated FRFTLFM synchronization framework and sequence optimization scheme are rigorously evaluated via comprehensive simulations.The results demonstrate substantial improvements in synchronization accuracy and computational efficiency compared with conventional methods,particularly under time-frequency dual selective fading LEO satellite channels.The algorithm provides a robust and reliable solution for time-frequency synchronization in LEO satellite communication systems,thereby enhancing overall system performance and reliability.展开更多
The image-based approach is widely used in fault detection(FD)algorithms of mechanical systems.The images are derived from the vibrational signals transformed from the time to time–frequency domain,and they are used ...The image-based approach is widely used in fault detection(FD)algorithms of mechanical systems.The images are derived from the vibrational signals transformed from the time to time–frequency domain,and they are used to develop a convolutional neural network(CNN)to automate the FD process.Nowadays,images are also obtained from the transformation of vibrational signals from the time domain to symmetrized dot pattern(SDP)coordinates,achieving high CNN testing accuracy.This paper shows a comparison of image-CNN approaches for FD using images obtained from time–frequency transforms and those obtained from the SDP transform as input.The comparison was conducted using experimental data from two publicly available bearing datasets,examining both the accuracy of the CNNs and the computational time required for the vibrational signal transformations.The results show that the SDP-CNN approach achieves the same accuracy as spectrogram-CNN approaches but with a significantly reduced computational time.These results support the future real-time implementation of the SDPCNN approach for FD in mechanical systems such as bearings.展开更多
A dynamic graph(DG)is adopted to portray the evolving interplay between nodes in real-world scenarios prevalently.A high-order graph convolutional network(HGCN)is equipped with the ability to represent a DG by the spa...A dynamic graph(DG)is adopted to portray the evolving interplay between nodes in real-world scenarios prevalently.A high-order graph convolutional network(HGCN)is equipped with the ability to represent a DG by the spatial-temporal message passing mechanism built on tensor product.Concretely,an HGCN utilizes the discrete Fourier transform(DFT)to implement temporal message passing and then employs face-wise product to realize spatial message passing.However,DFT is only a special case of assorted time-frequency transforms,which considers the complex temporal patterns partially,thereby resulting in an inaccurate temporal message passing possibly.To address this issue,this study proposes six advanced time-frequency transform-incorporated HGCNs(TF-HGCNs)with discrete Fourier,discrete Hartley,discrete cosine,Haar wavelet,Walsh Hadamard,and slant transforms.In addition,a potent ensemble is built regarding the proposed six TF-HGCNs as the bases.Finally,the corresponding theoretical proof is presented.Empirical studies on six DG datasets demonstrate that owing to diverse time-frequency transforms,the proposed six TF-HGCNs significantly outperform state-of-the-art models in addressing the task of link weight estimation.Moreover,their ensemble outstrips each base's performance.展开更多
Airport disruptions often pose challenges in assigning aircraft to gates,resulting in infeasible planned schedules.In particular,a large number of transfer passengers miss their connections in the context of disruptio...Airport disruptions often pose challenges in assigning aircraft to gates,resulting in infeasible planned schedules.In particular,a large number of transfer passengers miss their connections in the context of disruptions,which cause huge economic losses to airlines and serious passengers’dissatisfaction.This paper proposes a set-partitioning-based model to optimize Aircraft-Gate Reassignment with Transfer Passenger Connections(AGRP-TPC),which incorporates flexible gate-swap and aircraft-delay operations to mitigate the overall impact of disruptions.To efficiently solve the model,we introduce the concepts of additive-transfer and nonstop-transfer to handle passenger connections,and develop a Hierarchical Column-and-Row Generation(HCRG)approach guided by airport terminal space attribute.The column generation and row generation procedures solve iteratively until no new variables and constraints are generated.In addition,a follow-on strategy and a diving heuristic are designed to efficiently obtain high-quality solutions.We evaluate the proposed approach using various instances from a major Chinese international airport.Computational results demonstrate that our approach outperforms the comparison algorithms and produces good solutions within the time limit.Detailed results indicate that our approach effectively reduces overall losses in aircraft-gate reassignment following disruptions,and it can serve as an auxiliary decision-making tool for airport operators and airlines.展开更多
With the increasingly complex and changeable electromagnetic environment,wireless communication systems are facing jamming and abnormal signal injection,which significantly affects the normal operation of a communicat...With the increasingly complex and changeable electromagnetic environment,wireless communication systems are facing jamming and abnormal signal injection,which significantly affects the normal operation of a communication system.In particular,the abnormal signals may emulate the normal signals,which makes it very challenging for abnormal signal recognition.In this paper,we propose a new abnormal signal recognition scheme,which combines time-frequency analysis with deep learning to effectively identify synthetic abnormal communication signals.Firstly,we emulate synthetic abnormal communication signals including seven jamming patterns.Then,we model an abnormal communication signals recognition system based on the communication protocol between the transmitter and the receiver.To improve the performance,we convert the original signal into the time-frequency spectrogram to develop an image classification algorithm.Simulation results demonstrate that the proposed method can effectively recognize the abnormal signals under various parameter configurations,even under low signal-to-noise ratio(SNR)and low jamming-to-signal ratio(JSR)conditions.展开更多
Two novel diketopiperazines(1 and 5),along with ten known compounds(2−4,6−12)demonstrating significant skin inflammation inhibition,were isolated from a marine-derived fungus identified as Aspergillus sp.FAZW0001.The ...Two novel diketopiperazines(1 and 5),along with ten known compounds(2−4,6−12)demonstrating significant skin inflammation inhibition,were isolated from a marine-derived fungus identified as Aspergillus sp.FAZW0001.The structural elucidation and configurational reassessments of compounds 1−5 were established through comprehensive spectral analyses,with their absolute configurations determined via single crystal X-ray diffraction using Cu Kαradiation,Marfey’s method,and comparison between experimental and calculated electronic circular dichroism(ECD)spectra.Compounds 1,2,and 8 exhibited significant anti-inflammatory activities in Propionibacterium acnes(P.acnes)-induced human monocyte cell lines.Compound 8 demonstrated the ability to down-regulate interleukin-1β(IL-1β)expression by inhibiting Toll-like receptor 2(TLR2)expression and modulating the activation of myeloid differentiation factor 88(MyD88),mitogen-activated protein kinase(MAPK),and nuclear factorκB(NF-κB)signaling pathways,thus reducing the cellular inflammatory response induced by P.acnes.Additionally,compound 8 showed the capacity to suppress mitochondrial reactive oxygen species(ROS)production and nucleotide-binding oligomerization domain-like receptor protein 3(NLRP3)inflammasome activation,thereby reducing IL-1βmaturation and secretion.A three-dimensional quantitative structure-activity relationships(3D-QSAR)model was applied to compounds 5−12 to analyze their anti-inflammatory structure-activity relationships.展开更多
A time-frequency signal processing method for two-phase flow through a horizontal Venturi based on adaptive optimal-kernel (AOK) was presented in this paper.First,the collected dynamic differential pressure signal o...A time-frequency signal processing method for two-phase flow through a horizontal Venturi based on adaptive optimal-kernel (AOK) was presented in this paper.First,the collected dynamic differential pressure signal of gas-liquid two-phase flow was preprocessed,and then the AOK theory was used to analyze the dynamic differ-ential pressure signal.The mechanism of two-phase flow was discussed through the time-frequency spectrum.On the condition of steady water flow rate,with the increasing of gas flow rate,the flow pattern changes from bubbly flow to slug flow,then to plug flow,meanwhile,the energy distribution of signal fluctuations show significant change that energy transfer from 15-35 Hz band to 0-8 Hz band;moreover,when the flow pattern is slug flow,there are two wave peaks showed in the time-frequency spectrum.Finally,a number of characteristic variables were defined by using the time-frequency spectrum and the ridge of AOK.When the characteristic variables were visu-ally analyzed,the relationship between different combination of characteristic variables and flow patterns would be gotten.The results show that,this method can explain the law of flow in different flow patterns.And characteristic variables,defined by this method,can get a clear description of the flow information.This method provides a new way for the flow pattern identification,and the percentage of correct prediction is up to 91.11%.展开更多
High-precision and real-time diagnosis of sucker rod pumping system(SRPS)is important for quickly mastering oil well operations.Deep learning-based method for classifying the dynamometer card(DC)of oil wells is an eff...High-precision and real-time diagnosis of sucker rod pumping system(SRPS)is important for quickly mastering oil well operations.Deep learning-based method for classifying the dynamometer card(DC)of oil wells is an efficient diagnosis method.However,the input of the DC as a two-dimensional image into the deep learning framework suffers from low feature utilization and high computational effort.Additionally,different SRPSs in an oil field have various system parameters,and the same SRPS generates different DCs at different moments.Thus,there is heterogeneity in field data,which can dramatically impair the diagnostic accuracy.To solve the above problems,a working condition recognition method based on 4-segment time-frequency signature matrix(4S-TFSM)and deep learning is presented in this paper.First,the 4-segment time-frequency signature(4S-TFS)method that can reduce the computing power requirements is proposed for feature extraction of DC data.Subsequently,the 4S-TFSM is constructed by relative normalization and matrix calculation to synthesize the features of multiple data and solve the problem of data heterogeneity.Finally,a convolutional neural network(CNN),one of the deep learning frameworks,is used to determine the functioning conditions based on the 4S-TFSM.Experiments on field data verify that the proposed diagnostic method based on 4S-TFSM and CNN(4S-TFSM-CNN)can significantly improve the accuracy of working condition recognition with lower computational cost.To the best of our knowledge,this is the first work to discuss the effect of data heterogeneity on the working condition recognition performance of SRPS.展开更多
Resource allocation is an important problem influencing the service quality of multi-beam satellite communications.In multi-beam satellite communications, the available frequency bandwidth is limited, users requiremen...Resource allocation is an important problem influencing the service quality of multi-beam satellite communications.In multi-beam satellite communications, the available frequency bandwidth is limited, users requirements vary rapidly, high service quality and joint allocation of multi-dimensional resources such as time and frequency are required. It is a difficult problem needs to be researched urgently for multi-beam satellite communications, how to obtain a higher comprehensive utilization rate of multidimensional resources, maximize the number of users and system throughput, and meet the demand of rapid allocation adapting dynamic changed the number of users under the condition of limited resources, with using an efficient and fast resource allocation algorithm.In order to solve the multi-dimensional resource allocation problem of multi-beam satellite communications, this paper establishes a multi-objective optimization model based on the maximum the number of users and system throughput joint optimization goal, and proposes a multi-objective deep reinforcement learning based time-frequency two-dimensional resource allocation(MODRL-TF) algorithm to adapt dynamic changed the number of users and the timeliness requirements. Simulation results show that the proposed algorithm could provide higher comprehensive utilization rate of multi-dimensional resources,and could achieve multi-objective joint optimization,and could obtain better timeliness than traditional heuristic algorithms, such as genetic algorithm(GA)and ant colony optimization algorithm(ACO).展开更多
In this study, several advanced analysis methods are applied to understand the relationships between the Nino-3.4 sea surface temperatures (SST) and the SSTs related to the tropical Indian Ocean Dipole (IOD). By a...In this study, several advanced analysis methods are applied to understand the relationships between the Nino-3.4 sea surface temperatures (SST) and the SSTs related to the tropical Indian Ocean Dipole (IOD). By analyzing a long data record, the authors focus on the time-frequency characteristics of these relationships, and of the structure of IOD. They also focus on the seasonal dependence of those characteristics in both time and frequency domains. Among the Nino-3.4 SST, IOD, and SSTs over the tropical western Indian Ocean (WIO) and eastern Indian Ocean (EIO), the WIO SST has the strongest annual and semiannual oscillations. While the Nino-3.4 SST has large inter-annual variability that is only second to its annual variability, the IOD is characterized by the largest semiannual oscillation, which is even stronger than its annual oscillation. The IOD is strongly and stably related to the EIO SST in a wide range of frequency bands and in all seasons. However, it is less significantly related to the WIO SST in the boreal winter and spring. There exists a generally weak and unstable relationship between the WIO and EIO SSTs, especially in the biennial and higher frequency bands. The relationship is especially weak in summer and fall, when IOD is apparent, but appears highly positive in winter and spring, when the IOD is unimportantly weak and even disappears. This feature reflects a caution in the definition and application of IOD. The Nino-3.4 SST has a strong positive relationship with the WIO SST in all seasons, mainly in the biennial and longer frequency bands. However, it shows no significant relationship with the EIO SST in summer and fall, and with IOD in winter and spring.展开更多
The interrupted sampling repeater jamming(ISRJ) is an effective deception jamming method for coherent radar, especially for the wideband linear frequency modulation(LFM) radar. An electronic counter-countermeasure...The interrupted sampling repeater jamming(ISRJ) is an effective deception jamming method for coherent radar, especially for the wideband linear frequency modulation(LFM) radar. An electronic counter-countermeasure(ECCM) scheme is proposed to remove the ISRJ-based false targets from the pulse compression result of the de-chirping radar. Through the time-frequency(TF) analysis of the radar echo signal, it can be found that the TF characteristics of the ISRJ signal are discontinuous in the pulse duration because the ISRJ jammer needs short durations to receive the radar signal. Based on the discontinuous characteristics a particular band-pass filter can be generated by two alternative approaches to retain the true target signal and suppress the ISRJ signal. The simulation results prove the validity of the proposed ECCM scheme for the ISRJ.展开更多
Predicting the time-varying auto-spectral density of a spacecraft in high-altitude orbits requires an accurate model for the non-stationary random vibration signals with densely spaced modal frequency. The traditional...Predicting the time-varying auto-spectral density of a spacecraft in high-altitude orbits requires an accurate model for the non-stationary random vibration signals with densely spaced modal frequency. The traditional time-varying algorithm limits prediction accuracy, thus affecting a number of operational decisions. To solve this problem, a time-varying auto regressive (TVAR) model based on the process neural network (PNN) and the empirical mode decomposition (EMD) is proposed. The time-varying system is tracked on-line by establishing a time-varying parameter model, and then the relevant parameter spectrum is obtained. Firstly, the EMD method is utilized to decompose the signal into several intrinsic mode functions (IMFs). Then for each IMF, the PNN is established and the time-varying auto-spectral density is obtained. Finally, the time-frequency distribution of the signals can be reconstructed by linear superposition. The simulation and the analytical results from an example demonstrate that this approach possesses simplicity, effectiveness, and feasibility, as well as higher frequency resolution.展开更多
The aim of this paper is to present a new method for flight flutter modal parameter identification in noisy environment. This method employs a time-frequency (TF) filter to reduce the noise before identification, wh...The aim of this paper is to present a new method for flight flutter modal parameter identification in noisy environment. This method employs a time-frequency (TF) filter to reduce the noise before identification, which depends on the localization property of sweep excitation in TF domain. Then, a generalized total least square (GTLS) identification algorithm based on stochastic framework is applied to the enhanced data. System identification with noisy data is transformed into a generalized total least square problem, and the solution is carried out by the generalized singular value decomposition (GSVD) to avoid the intensive nonlinear optimization computation. A nearly maximum likelihood property can be achieved by 'optimally' weighted generalized total least square. Finally, the efficiency of the method is illustrated by means of flight test data.展开更多
Li transient concentration distribution in spherical active material particles can affect the maximum power density and the safe operating regime of the electric vehicles(EVs). On one hand, the quasiexact/exact soluti...Li transient concentration distribution in spherical active material particles can affect the maximum power density and the safe operating regime of the electric vehicles(EVs). On one hand, the quasiexact/exact solution obtained in the time/frequency domain is time-consuming and just as a reference value for approximate solutions;on the other hand, calculation errors and application range of approximate solutions not only rely on approximate algorithms but also on discharge modes. For the purpose to track the transient dynamics for Li solid-phase diffusion in spherical active particles with a tolerable error range and for a wide applicable range, it is necessary to choose optimal approximate algorithms in terms of discharge modes and the nature of active material particles. In this study, approximation methods,such as diffusion length method, polynomial profile approximation method, Padé approximation method,pseudo steady state method, eigenfunction-based Galerkin collocation method, and separation of variables method for solving Li solid-phase diffusion in spherical active particles are compared from calculation fundamentals to algorithm implementation. Furthermore, these approximate solutions are quantitatively compared to the quasi-exact/exact solution in the time/frequency domain under typical discharge modes, i.e., start-up, slow-down, and speed-up. The results obtained from the viewpoint of time-frequency analysis offer a theoretical foundation on how to track Li transient concentration profile in spherical active particles with a high precision and for a wide application range. In turn, optimal solutions of Li solid diffusion equations for spherical active particles can improve the reliability in predicting safe operating regime and estimating maximum power for automotive batteries.展开更多
基金supported by the National Natural Science Foundation of China(No.62472118)the Guangxi Science and Technology Program(No.AB24010315)+2 种基金the Central Guidance on Local Science and Technology Development Fund of Guangxi Province(No.ZY23055008)the Innovation Project of Guangxi Graduate Education(No.YCSW2025348)the Innovation Platform and Talent Program of Guilin City(No.20220124-12).
摘要With the rapid development of Artificial Intelligence of Things(AIoT)technologies,the security of Industrial Internet of Things(IIoT)data faces increasing challenges,particularly in time series anomaly detection.IIoT data are typically scarce in abnormal samples and noisy,making unsupervised learning a common solution.The security challenges of IIoT data in AIoT environments require robust unsupervised anomaly detection methods.While Variational Autoencoders(VAEs)excel in noise resilience,they face two critical challenges in IIoT data:difficulties in single-variable time-series modeling and conflicts between static prior assumptions and dynamic temporal features.To address these challenges,we propose the Greater Cane Rat Algorithm-enhanced FourierWavelet Conditional Variational Autoencoder(GCRA-FWVAE).Our method introduces a time-frequency dualbranch architecture that synergistically combines wavelet transforms for localized transient feature extraction and Fourier transforms for global spectral characterization.These complementary representations jointly regulate the Conditional Variational Autoencoder(CVAE)reconstruction process,effectively preserving critical anomaly signatures while suppressing noise interference.The architecture is further optimized through bioinspired Greater Cane Rat Algorithm(GCRA)to improve adaptive learning capabilities.Extensive validation on the Yahoo benchmark indicates state-of-the-art performance,achieving an F1-score of 93.6%(an improvement of 4.5% over baseline VAEs)and a precision of 95.1%.These improvements significantly increase anomaly detection accuracy and robustness,particularly in the AIoT environment,where it effectively handles more complex and dynamic industrial data.
基金supported by the Innovative Human Resource Development for Local Intel-lectualization program through the Institute of Information&Communications Technology Planning&Evaluation(IITP)grant funded by the Korea government(MSIT)(No.IITP-2026-2020-0-01741)the research fund of Hanyang University(HY-2025-1110).
摘要Arrhythmias are a frequently occurring phenomenon in clinical practice,but how to accurately dis-tinguish subtle rhythm abnormalities remains an ongoing difficulty faced by the entire research community when conducting ECG-based studies.From a review of existing studies,two main factors appear to contribute to this problem:the uneven distribution of arrhythmia classes and the limited expressiveness of features learned by current models.To overcome these limitations,this study proposes a dual-path multimodal framework,termed DM-EHC(Dual-Path Multimodal ECG Heartbeat Classifier),for ECG-based heartbeat classification.The proposed framework links 1D ECG temporal features with 2D time–frequency features.By setting up the dual paths described above,the model can process more dimensions of feature information.The MIT-BIH arrhythmia database was selected as the baseline dataset for the experiments.Experimental results show that the proposed method outperforms single modalities and performs better for certain specific types of arrhythmias.The model achieved mean precision,recall,and F1 score of 95.14%,92.26%,and 93.65%,respectively.These results indicate that the framework is robust and has potential value in automated arrhythmia classification.
基金supported by the National Natural Science Foundation of China(Grant Nos.52379098,52274075)the Project of Xingliao Talents Program(XLYC2203008)the Science and Technology Program Project of Liaoning Province(2025JH2/101900011).
摘要Understanding how rock slopes respond to blasting loads is crucial for maintaining excavation safety and slope stability.Nevertheless,the spatiotemporal evolution,nonlinear dependence on blasting parameters,and predictive behavior of dominant frequency responses in slope vibrations remain insufficiently understood and quantified.This study combines time-frequency analysis with machine learning to explore how the dominant frequency(fd)evolves in slopes under blasting.Continuous Wavelet Transform(CWT)was employed to characterize the temporal-frequency evolution of vibration signals,revealing that the dominant frequency exhibits strong spatial dependence and nonlinear variability influenced by blasting parameters and rock mass structures.Three machine learning models,namely Back Propagation Neural Network(BP),Support Vector Machine(SVM),and Random Forest(RF),were developed to predict fd based on 1,000 monitoring samples obtained from numerical and field simulations.Among them,the RF model achieved the highest prediction accuracy,with mean absolute percentage errors(MAPE)below 15%,demonstrating strong robustness and generalization capability.Our analysis shows that external excitation factors,especially the loading frequency(fd),mainly control the frequency response,while internal controlling factors,such as spatial position,lithological variation,and mechanical heterogeneity,modulate localized frequency amplification and energy redistribution.The results reveal that fd tends to decrease with elevation and distance from the blasting source,whereas structural planes and weathered zones induce high-frequency amplification due to scattering and modal coupling effects.This study offers a new framework combining time-frequency analysis and machine learning to measure the nonlinear interaction between blasting and rock mass response,offering new insights for dynamic stability evaluation and hazard mitigation in complex rock slope systems.
基金National Natural Science Foundation of China(Grant Nos.42388102,42030105,42192535)the Open Fund of State Key Laboratory of Precision Geodesy,Innovation Academy for Precision Measurement Science and Technology,Chinese Academy of Sciences(Grant No.SKLPG2025-1-5)。
摘要The state-of-the-art optical atomic clocks and the time-frequency signal transmission open a fresh field for gravity potential(geopotential)determination.Various methods,including optical fiber frequency transfer,satellite two-way,satellite common-view,satellite carrier phase,VLBI,tri-frequency combination,and dual-frequency combination,were developed to determine the geopotential differences using optical atomic clocks and then determine the geopotential at station B based on the geopotential at station A.This review elaborates the principles,methods,scientific objectives,applications,and relevant research trends of geopotential determination based on time-frequency signals.
基金funded in part by The National Natural Science Foundation of China(No.62161006)Guangxi Science and Technology Programunder Grant No.FN2504240022Innovation Project of GUET Graduate Education(No.2025YCXS078).
摘要In recent years,anomaly detection in Wireless Sensor Networks(WSNs)has been widely studied using Graph Neural Networks and Transformer-based methods.However,in multi-node and multi-modal data scenarios,these approaches still face challenges such as insufficient extraction of spatiotemporal correlation features,limited modeling capabilities when relying solely on either time-domain or frequency-domain information,and high computational overhead.To address these issues,this work aims to develop an anomaly detection model that balances detection performance with computational efficiency,enabling effective identification of complex anomaly patterns.Specifically,we propose a time–frequency feature extraction method with topological information enhancement,topology-enhanced multi-modal spatio-temporal anomaly detection(TE-MSTAD).Building upon the Receptance Weighted Key Value(RWKV)model with linear complexity,a cross-modal feature extraction module is introduced to strengthen the modeling of multi-modal correlations.Meanwhile,adaptive adjacency matrices are constructed by integrating time–frequency features and combining outputs from different Graph Neural Networks,thereby enhancing topological information.Furthermore,a dual-branch structure is designed to jointly model time-domain and frequency-domain features,improving the extraction of complex anomaly characteristics.Experiments on both publicly available datasets and real-world collected data demonstrate that the proposed method achieves F1-scores of 92.52%and 93.28%,respectively,outperforming existing methods in detection performance and generalization capability.
基金Supported by National Natural Science Foundation of China(Grant No.62271230)Shandong Provincial Central Guidance on Local Science and Technology Development Fund(Grant No.YDZX2022178).
摘要Fault features in mechanical systems often manifest as transient impulses,which can be effectively analyzed using time-frequency analysis(TFA)methods.Recently,a new TFA technique known as the time-reassigned multi-synchrosqueezing transform(TMssT)was proposed to capture these transient impulses for fault diagnosis.However,the TMSST,which is based on the short-time Fourier transform(STFT),suffers from unclear high-frequency re-presentations owing to the fixed sliding window used in the STFT.To address this limitation,the current study combined TMSST with the S-transform and a local maximum method to enhance the time-frequency representation for improved signal analysis.Furthermore,an extractive reconstruction algorithm that binds the maximum value of the spectral envelope is proposed for spectral decomposition.To validate the proposed technique,a simulated noise-added signal and four experimental bearing defect datasets were used.The results demonstrate that the proposed technique can effectively and accurately extract fault features from bearing signals regardless of whether the bearings operate under constant or varying speed conditions.This study offers a novel and efficient approach for fault diagnosis in mechanical systems with complex dynamic behaviors.
基金supported by the Beijing Natural Science Foundation(4252008)the Natural Science Foundation of Chongqing Province(CSTB2024NSCQLZX0176)the Beijing Natural Science Foundation of Undergraduate Qiyan Program(QY24197)。
摘要With the development of Sixth-Generation(6G)mobile communication technologies,Low Earth Orbit(LEO)satellite communication systems have become extremely important in mobile communications owing to their large coverage,high efficiency,and low cost.However,the high dynamic LEO satellite channels cause serious time-frequency dual selective fading,significantly impairing the performance of conventional single time or frequency domain synchronization algorithms and limiting their applicability.To address these challenges,this paper proposes a synchronization algorithm based on Linear Frequency Modulation(LFM)signals and the Fractional Fourier Transform(FRFT).Exploiting the inherent robustness of LFM signals against frequency deviations and multipath effects,coupled with their energy concentration property in the optimal fractional Fourier domain,the proposed algorithm enables efficient synchronization with enhanced resilience to time-frequency variations.Furthermore,LFM preamble sequences are optimally designed for diverse channel conditions.This work presents a theoretical analysis of the time-frequency nonstationary characteristics of LEO satellite channels and discusses the performance limitations of traditional synchronization algorithms.The proposed integrated FRFTLFM synchronization framework and sequence optimization scheme are rigorously evaluated via comprehensive simulations.The results demonstrate substantial improvements in synchronization accuracy and computational efficiency compared with conventional methods,particularly under time-frequency dual selective fading LEO satellite channels.The algorithm provides a robust and reliable solution for time-frequency synchronization in LEO satellite communication systems,thereby enhancing overall system performance and reliability.
摘要The image-based approach is widely used in fault detection(FD)algorithms of mechanical systems.The images are derived from the vibrational signals transformed from the time to time–frequency domain,and they are used to develop a convolutional neural network(CNN)to automate the FD process.Nowadays,images are also obtained from the transformation of vibrational signals from the time domain to symmetrized dot pattern(SDP)coordinates,achieving high CNN testing accuracy.This paper shows a comparison of image-CNN approaches for FD using images obtained from time–frequency transforms and those obtained from the SDP transform as input.The comparison was conducted using experimental data from two publicly available bearing datasets,examining both the accuracy of the CNNs and the computational time required for the vibrational signal transformations.The results show that the SDP-CNN approach achieves the same accuracy as spectrogram-CNN approaches but with a significantly reduced computational time.These results support the future real-time implementation of the SDPCNN approach for FD in mechanical systems such as bearings.
基金supported in part by the National Natural Science Foundation of China(62372385,62272078,62002337)Chongqing Natural Science Foundation(CSTB2022NSCQ-MSX1486,CSTB2023NSCQ-LZX0069)。
摘要A dynamic graph(DG)is adopted to portray the evolving interplay between nodes in real-world scenarios prevalently.A high-order graph convolutional network(HGCN)is equipped with the ability to represent a DG by the spatial-temporal message passing mechanism built on tensor product.Concretely,an HGCN utilizes the discrete Fourier transform(DFT)to implement temporal message passing and then employs face-wise product to realize spatial message passing.However,DFT is only a special case of assorted time-frequency transforms,which considers the complex temporal patterns partially,thereby resulting in an inaccurate temporal message passing possibly.To address this issue,this study proposes six advanced time-frequency transform-incorporated HGCNs(TF-HGCNs)with discrete Fourier,discrete Hartley,discrete cosine,Haar wavelet,Walsh Hadamard,and slant transforms.In addition,a potent ensemble is built regarding the proposed six TF-HGCNs as the bases.Finally,the corresponding theoretical proof is presented.Empirical studies on six DG datasets demonstrate that owing to diverse time-frequency transforms,the proposed six TF-HGCNs significantly outperform state-of-the-art models in addressing the task of link weight estimation.Moreover,their ensemble outstrips each base's performance.
基金supported by the National Natural Science Foundation of China(No.U2333218).
摘要Airport disruptions often pose challenges in assigning aircraft to gates,resulting in infeasible planned schedules.In particular,a large number of transfer passengers miss their connections in the context of disruptions,which cause huge economic losses to airlines and serious passengers’dissatisfaction.This paper proposes a set-partitioning-based model to optimize Aircraft-Gate Reassignment with Transfer Passenger Connections(AGRP-TPC),which incorporates flexible gate-swap and aircraft-delay operations to mitigate the overall impact of disruptions.To efficiently solve the model,we introduce the concepts of additive-transfer and nonstop-transfer to handle passenger connections,and develop a Hierarchical Column-and-Row Generation(HCRG)approach guided by airport terminal space attribute.The column generation and row generation procedures solve iteratively until no new variables and constraints are generated.In addition,a follow-on strategy and a diving heuristic are designed to efficiently obtain high-quality solutions.We evaluate the proposed approach using various instances from a major Chinese international airport.Computational results demonstrate that our approach outperforms the comparison algorithms and produces good solutions within the time limit.Detailed results indicate that our approach effectively reduces overall losses in aircraft-gate reassignment following disruptions,and it can serve as an auxiliary decision-making tool for airport operators and airlines.
基金supported by Natural Science Foundation of China(No.62371231)Natural Science Foundation on Frontier Leading Technology Basic Research Project of Jiangsu under Grant BK20222001Jiangsu Provincial Key Research and Development Program(No.BE2023027).
摘要With the increasingly complex and changeable electromagnetic environment,wireless communication systems are facing jamming and abnormal signal injection,which significantly affects the normal operation of a communication system.In particular,the abnormal signals may emulate the normal signals,which makes it very challenging for abnormal signal recognition.In this paper,we propose a new abnormal signal recognition scheme,which combines time-frequency analysis with deep learning to effectively identify synthetic abnormal communication signals.Firstly,we emulate synthetic abnormal communication signals including seven jamming patterns.Then,we model an abnormal communication signals recognition system based on the communication protocol between the transmitter and the receiver.To improve the performance,we convert the original signal into the time-frequency spectrogram to develop an image classification algorithm.Simulation results demonstrate that the proposed method can effectively recognize the abnormal signals under various parameter configurations,even under low signal-to-noise ratio(SNR)and low jamming-to-signal ratio(JSR)conditions.
基金supported by the National Natural Science Foundation of China(Nos.41876189 and 81703388)the State Key Laboratory of Bioreactor Engineering and Shanghai Collaborative Innovation Center for Biomanufacturing Technology.
摘要Two novel diketopiperazines(1 and 5),along with ten known compounds(2−4,6−12)demonstrating significant skin inflammation inhibition,were isolated from a marine-derived fungus identified as Aspergillus sp.FAZW0001.The structural elucidation and configurational reassessments of compounds 1−5 were established through comprehensive spectral analyses,with their absolute configurations determined via single crystal X-ray diffraction using Cu Kαradiation,Marfey’s method,and comparison between experimental and calculated electronic circular dichroism(ECD)spectra.Compounds 1,2,and 8 exhibited significant anti-inflammatory activities in Propionibacterium acnes(P.acnes)-induced human monocyte cell lines.Compound 8 demonstrated the ability to down-regulate interleukin-1β(IL-1β)expression by inhibiting Toll-like receptor 2(TLR2)expression and modulating the activation of myeloid differentiation factor 88(MyD88),mitogen-activated protein kinase(MAPK),and nuclear factorκB(NF-κB)signaling pathways,thus reducing the cellular inflammatory response induced by P.acnes.Additionally,compound 8 showed the capacity to suppress mitochondrial reactive oxygen species(ROS)production and nucleotide-binding oligomerization domain-like receptor protein 3(NLRP3)inflammasome activation,thereby reducing IL-1βmaturation and secretion.A three-dimensional quantitative structure-activity relationships(3D-QSAR)model was applied to compounds 5−12 to analyze their anti-inflammatory structure-activity relationships.
基金Supported by the Natural Science Foundation of Zhejiang Province(Y1100842) the Planning Projects of General Administration of Quality Supervision Inspection and Quarantine of the People's Republic of China(2006QK23)
摘要A time-frequency signal processing method for two-phase flow through a horizontal Venturi based on adaptive optimal-kernel (AOK) was presented in this paper.First,the collected dynamic differential pressure signal of gas-liquid two-phase flow was preprocessed,and then the AOK theory was used to analyze the dynamic differ-ential pressure signal.The mechanism of two-phase flow was discussed through the time-frequency spectrum.On the condition of steady water flow rate,with the increasing of gas flow rate,the flow pattern changes from bubbly flow to slug flow,then to plug flow,meanwhile,the energy distribution of signal fluctuations show significant change that energy transfer from 15-35 Hz band to 0-8 Hz band;moreover,when the flow pattern is slug flow,there are two wave peaks showed in the time-frequency spectrum.Finally,a number of characteristic variables were defined by using the time-frequency spectrum and the ridge of AOK.When the characteristic variables were visu-ally analyzed,the relationship between different combination of characteristic variables and flow patterns would be gotten.The results show that,this method can explain the law of flow in different flow patterns.And characteristic variables,defined by this method,can get a clear description of the flow information.This method provides a new way for the flow pattern identification,and the percentage of correct prediction is up to 91.11%.
基金We would like to thank the associate editor and the reviewers for their constructive comments.This work was supported in part by the National Natural Science Foundation of China under Grant 62203234in part by the State Key Laboratory of Robotics of China under Grant 2023-Z03+1 种基金in part by the Natural Science Foundation of Liaoning Province under Grant 2023-BS-025in part by the Research Program of Liaoning Liaohe Laboratory under Grant LLL23ZZ-02-02.
摘要High-precision and real-time diagnosis of sucker rod pumping system(SRPS)is important for quickly mastering oil well operations.Deep learning-based method for classifying the dynamometer card(DC)of oil wells is an efficient diagnosis method.However,the input of the DC as a two-dimensional image into the deep learning framework suffers from low feature utilization and high computational effort.Additionally,different SRPSs in an oil field have various system parameters,and the same SRPS generates different DCs at different moments.Thus,there is heterogeneity in field data,which can dramatically impair the diagnostic accuracy.To solve the above problems,a working condition recognition method based on 4-segment time-frequency signature matrix(4S-TFSM)and deep learning is presented in this paper.First,the 4-segment time-frequency signature(4S-TFS)method that can reduce the computing power requirements is proposed for feature extraction of DC data.Subsequently,the 4S-TFSM is constructed by relative normalization and matrix calculation to synthesize the features of multiple data and solve the problem of data heterogeneity.Finally,a convolutional neural network(CNN),one of the deep learning frameworks,is used to determine the functioning conditions based on the 4S-TFSM.Experiments on field data verify that the proposed diagnostic method based on 4S-TFSM and CNN(4S-TFSM-CNN)can significantly improve the accuracy of working condition recognition with lower computational cost.To the best of our knowledge,this is the first work to discuss the effect of data heterogeneity on the working condition recognition performance of SRPS.
基金supported by the National Key Research and Development Program of China under No. 2019YFB1803200。
摘要Resource allocation is an important problem influencing the service quality of multi-beam satellite communications.In multi-beam satellite communications, the available frequency bandwidth is limited, users requirements vary rapidly, high service quality and joint allocation of multi-dimensional resources such as time and frequency are required. It is a difficult problem needs to be researched urgently for multi-beam satellite communications, how to obtain a higher comprehensive utilization rate of multidimensional resources, maximize the number of users and system throughput, and meet the demand of rapid allocation adapting dynamic changed the number of users under the condition of limited resources, with using an efficient and fast resource allocation algorithm.In order to solve the multi-dimensional resource allocation problem of multi-beam satellite communications, this paper establishes a multi-objective optimization model based on the maximum the number of users and system throughput joint optimization goal, and proposes a multi-objective deep reinforcement learning based time-frequency two-dimensional resource allocation(MODRL-TF) algorithm to adapt dynamic changed the number of users and the timeliness requirements. Simulation results show that the proposed algorithm could provide higher comprehensive utilization rate of multi-dimensional resources,and could achieve multi-objective joint optimization,and could obtain better timeliness than traditional heuristic algorithms, such as genetic algorithm(GA)and ant colony optimization algorithm(ACO).
摘要In this study, several advanced analysis methods are applied to understand the relationships between the Nino-3.4 sea surface temperatures (SST) and the SSTs related to the tropical Indian Ocean Dipole (IOD). By analyzing a long data record, the authors focus on the time-frequency characteristics of these relationships, and of the structure of IOD. They also focus on the seasonal dependence of those characteristics in both time and frequency domains. Among the Nino-3.4 SST, IOD, and SSTs over the tropical western Indian Ocean (WIO) and eastern Indian Ocean (EIO), the WIO SST has the strongest annual and semiannual oscillations. While the Nino-3.4 SST has large inter-annual variability that is only second to its annual variability, the IOD is characterized by the largest semiannual oscillation, which is even stronger than its annual oscillation. The IOD is strongly and stably related to the EIO SST in a wide range of frequency bands and in all seasons. However, it is less significantly related to the WIO SST in the boreal winter and spring. There exists a generally weak and unstable relationship between the WIO and EIO SSTs, especially in the biennial and higher frequency bands. The relationship is especially weak in summer and fall, when IOD is apparent, but appears highly positive in winter and spring, when the IOD is unimportantly weak and even disappears. This feature reflects a caution in the definition and application of IOD. The Nino-3.4 SST has a strong positive relationship with the WIO SST in all seasons, mainly in the biennial and longer frequency bands. However, it shows no significant relationship with the EIO SST in summer and fall, and with IOD in winter and spring.
基金supported by the National Natural Science Foundation of China(61271442)
摘要The interrupted sampling repeater jamming(ISRJ) is an effective deception jamming method for coherent radar, especially for the wideband linear frequency modulation(LFM) radar. An electronic counter-countermeasure(ECCM) scheme is proposed to remove the ISRJ-based false targets from the pulse compression result of the de-chirping radar. Through the time-frequency(TF) analysis of the radar echo signal, it can be found that the TF characteristics of the ISRJ signal are discontinuous in the pulse duration because the ISRJ jammer needs short durations to receive the radar signal. Based on the discontinuous characteristics a particular band-pass filter can be generated by two alternative approaches to retain the true target signal and suppress the ISRJ signal. The simulation results prove the validity of the proposed ECCM scheme for the ISRJ.
基金Aeronautical Science Foundation of China (20071551016)
摘要Predicting the time-varying auto-spectral density of a spacecraft in high-altitude orbits requires an accurate model for the non-stationary random vibration signals with densely spaced modal frequency. The traditional time-varying algorithm limits prediction accuracy, thus affecting a number of operational decisions. To solve this problem, a time-varying auto regressive (TVAR) model based on the process neural network (PNN) and the empirical mode decomposition (EMD) is proposed. The time-varying system is tracked on-line by establishing a time-varying parameter model, and then the relevant parameter spectrum is obtained. Firstly, the EMD method is utilized to decompose the signal into several intrinsic mode functions (IMFs). Then for each IMF, the PNN is established and the time-varying auto-spectral density is obtained. Finally, the time-frequency distribution of the signals can be reconstructed by linear superposition. The simulation and the analytical results from an example demonstrate that this approach possesses simplicity, effectiveness, and feasibility, as well as higher frequency resolution.
摘要The aim of this paper is to present a new method for flight flutter modal parameter identification in noisy environment. This method employs a time-frequency (TF) filter to reduce the noise before identification, which depends on the localization property of sweep excitation in TF domain. Then, a generalized total least square (GTLS) identification algorithm based on stochastic framework is applied to the enhanced data. System identification with noisy data is transformed into a generalized total least square problem, and the solution is carried out by the generalized singular value decomposition (GSVD) to avoid the intensive nonlinear optimization computation. A nearly maximum likelihood property can be achieved by 'optimally' weighted generalized total least square. Finally, the efficiency of the method is illustrated by means of flight test data.
基金the financial support from the National Science Foundation of China(22078190 and 12002196)the National Key Research and Development Program of China(2020YFB1505802)。
摘要Li transient concentration distribution in spherical active material particles can affect the maximum power density and the safe operating regime of the electric vehicles(EVs). On one hand, the quasiexact/exact solution obtained in the time/frequency domain is time-consuming and just as a reference value for approximate solutions;on the other hand, calculation errors and application range of approximate solutions not only rely on approximate algorithms but also on discharge modes. For the purpose to track the transient dynamics for Li solid-phase diffusion in spherical active particles with a tolerable error range and for a wide applicable range, it is necessary to choose optimal approximate algorithms in terms of discharge modes and the nature of active material particles. In this study, approximation methods,such as diffusion length method, polynomial profile approximation method, Padé approximation method,pseudo steady state method, eigenfunction-based Galerkin collocation method, and separation of variables method for solving Li solid-phase diffusion in spherical active particles are compared from calculation fundamentals to algorithm implementation. Furthermore, these approximate solutions are quantitatively compared to the quasi-exact/exact solution in the time/frequency domain under typical discharge modes, i.e., start-up, slow-down, and speed-up. The results obtained from the viewpoint of time-frequency analysis offer a theoretical foundation on how to track Li transient concentration profile in spherical active particles with a high precision and for a wide application range. In turn, optimal solutions of Li solid diffusion equations for spherical active particles can improve the reliability in predicting safe operating regime and estimating maximum power for automotive batteries.