Adaptive Fourier Decomposition Research Articles

Sparse representations of approximation to identity via time–space fractional heat equations

In this paper, we investigate sparse representations of approximation to identity via time–space fractional heat equations: { ∂ t β u ( t , x ) = − ν ( − Δ ) α / 2 u ( t , x ) , ( t , x ) ∈ R + 1 + n ; u ( 0 , x ) = f ( x ) , x ∈ R n . Due to the time-fractional derivative, the semigroup property is invalid for the solutions u ( ⋅ , ⋅ ) to the above problem. This deficiency makes it difficult to verify the boundary vanishing condition of u ( ⋅ , ⋅ ) , which is essential for getting the sparse representations. We develop a new method to avoid using the semigroup property. The analogous results are obtained for stochastic time–space fractional heat equations. As an application, we apply the adaptive Fourier decomposition to establish sparse representations of the solutions to the concerned equations.

An AFD-based adaptive impulse response model of time series

In this paper, an adaptive impulse response model of time series is proposed based on adaptive Fourier decomposition (AFD), which characterizes the data-driven impulse response model with “energy-resolution” and is applied to the effect analysis of shocks on time series. First, according to the observed time series, the frequency points of impulse response function are estimated, and the approximate expression of impulse response function in frequency domain is given. Second, AFD is carried out on the approximate impulse response function level by level, and the poles and coefficients of basis function of each level are obtained adaptively by using the criterion of energy maximization. Third, when the goodness of fit is greater than a certain threshold, all decomposition levels are determined and the adaptive impulse response model is established. Each decomposition level depicts the energy-based resolution characteristics of the impulse response model. Based on gold price return and US dollar index return in daily frequency from 2017 to 2022, this paper carried out AFD-based effect analysis with energy-resolution year by year. Results show that the proposed AFD-based adaptive impulse response model in this paper can describe the persistence effect of shocks of gold price on the US dollar index, and further measure the plainness of linkage relationship between these two time series. Decomposing impulse response to each decomposition level reveals consistence, convergency, and sufficiency of the impulse response model. The AFD-based adaptive data-driven model provides a new way for the analysis of time series in time-frequency domain.

Trends by Adaptive Fourier Decomposition and Application in Prediction

Trends by Adaptive Fourier Decomposition and Application in Prediction

Fault Feature Extraction Method of Rolling Bearing Based on IAFD and TKEO

The study of bearing fault feature extraction using adaptive Fourier decomposition (AFD) holds significant practical importance. However, AFD is constrained by its reliance on prior knowledge for determining decomposition levels, which can result in either underdecomposition or overdecomposition based on a single indicator. Consequently, an improved adaptive Fourier decomposition (IAFD) is proposed. First, a combined weight index called SP is constructed, and the whale optimization algorithm is employed to optimize the SP weight parameter. Second, the IAFD decomposition levels can be adaptively determined using the optimized SP. Finally, a feature extraction method-based IAFD and Teager–Kaiser energy operator is applied in rolling bearing fault diagnosis. Case studies on the Case Western Reserve University and self-made KUST-SY datasets validate the effectiveness of the proposed method.

A novel image encryption scheme with adaptive Fourier decomposition

This study creatively introduces a class of adaptive Fourier decomposition (AFD) techniques into the field of image encryption, where AFD and its variants are newly developed signal decomposition methods. The novelty of this article is twofold. First, we use three different AFD techniques, including Core AFD, Unwinding AFD, and Cyclic AFD, to generate two key streams through decomposing a source image. The two key streams generated by different AFD algorithms are different and aperiodic, which can effectively improve the security of encryption algorithm. Furthermore, this work also proposes a novel encryption method that implements the confusion and diffusion operations by index-sorted mapping method and pixel swapping operation based on two generated key streams, respectively. To ensure that the image is fully scrambled, both pixel-level and bit-level permutations are performed, which employs a novel bilateral sequence diffusion method to change the image pixel distribution based on pixel swap operations. The cryptographic performance of the scheme is evaluated through various security analyses, including key sensitivity, statistical, entropy, clipping attack, and differential attack analysis. Experimental results confirm that the proposed image encryption scheme ensures a high level of security and exhibits superior performance in resisting various attacks compared with several traditional and state-of-the-art image encryption methods.

A Deep Stochastic Adaptive Fourier Decomposition Network for Hyperspectral Image Classification.

Deep learning-based hyperspectral image (HSI) classification methods have recently shown excellent performance, however, there are two shortcomings that need to be addressed. One is that deep network training requires a large number of labeled images, and the other is that deep network needs to learn a large number of parameters. They are also general problems of deep networks, especially in applications that require professional techniques to acquire and label images, such as HSI and medical images. In this paper, we propose a deep network architecture (SAFDNet) based on the stochastic adaptive Fourier decomposition (SAFD) theory. SAFD has powerful unsupervised feature extraction capabilities, so the entire deep network only requires a small number of annotated images to train the classifier. In addition, we use fewer convolution kernels in the entire deep network, which greatly reduces the number of deep network parameters. SAFD is a newly developed signal processing tool with solid mathematical foundation, which is used to construct the unsupervised deep feature extraction mechanism of SAFDNet. Experimental results on three popular HSI classification datasets show that our proposed SAFDNet outperforms other compared state-of-the-art deep learning methods in HSI classification.

Zero-Phase Filter-Based Adaptive Fourier Decomposition and Its Application to Fault Diagnosis of Rolling Bearing

Empirical wavelet transform (EWT) is a recently developed non-stationary signal decomposition approach. However, the number of spectrum divisions needs to be preset first, and the performance of spectrum division is inferior for a strong noisy or non-stationary signal caused by the mode mixing problem. To address this issue, a novel non-stationary signal decomposition method termed zero-phase filter-based adaptive Fourier decomposition (ZPF-AFD) is proposed in this paper. In the ZPF-AFD method, the number of spectrum divisions is adaptively determined first using the envelope entropy (EE) metric. Next, the spectral envelope processing (SEP) is applied to achieve an adaptive spectrum division. Last, the zero-phase filter (ZPF) is utilized to filter the frequency domain signal to obtain the components. The ZPF can effectively eliminate the mode mixing problem of EWT because of its no transition phase. The proposed ZPF-AFD approach is contrasted with the existing EMD, VMD, and EWT approaches through analyzing the simulated and measured signals in rolling bearing with local failures. The experimental results demonstrate that the proposed ZPF-AFD method has the best anti-noise and diagnosis accuracy, and its diagnosis performance is superior to the compared methods.

Daily fluctuations in COVID-19 infection rates under Tokyo's epidemic prevention measures - new evidence from adaptive Fourier decomposition.

The COVID-19 pandemic has witnessed widespread infections and variants. Particularly, Tokyo faced the challenge of seven waves of COVID-19, during which government interventions played a pivotal role. Therefore, gaining a comprehensive understanding of government control measures is of paramount importance, which is beneficial for health authorities in the policy development process. Our study analysis the daily change data of the daily COVID-19 infection count in Tokyo from January 16, 2020 to September 30, 2022. We utilized adaptive Fourier decomposition (AFD) for analyzing the temporal trends within COVID-19 data. It extends the conventional AFD approach by constructing new components base on multiple individual components at various time-frequency scales. Furthermore, we conducted Pearson correlation assessments of the first to third-order synthesis results, along with comparative analyses against other signal analysis techniques. Ultimately, these new components are integrated with policy data spanning different time periods for a comprehensive analysis. The analysis of daily COVID-19 data in Tokyo using AFD reveals how various government policies impacted infection rates across seven distinct fluctuation periods. In the decomposition results, the reduction of business hours policy correlated with high-frequency components in the first four waves, while the low-frequency components for the sixth wave suggested a decline in its relevance. The vaccination policy initially displayed a mid-frequency correlation with the fifth wave and continued with a low-frequency correlation in the last wave. Moreover, our statistical analysis (value of p < 0.05) demonstrated that 75% of the third-order AFD components exhibited significant positive correlations with the original infections, while the correlation coefficients of most components in EMD and VMD did not attain significance. In the time-frequency domain, AFD demonstrates superior performance compared to EMD and VMD in capturing crucial data related to epidemic control measures. The variations in daily COVID-19 infection counts during these seven periods under various policies are evident in distinct third-order AFD components. These findings guide the formulation of future public health policies and social measures.

Frequency analysis with multiple kernels and complete dictionary

In signal analysis, among the effort of seeking for efficient representations of a signal into the basic ones of meaningful frequencies, to extract principal frequency components, consecutively one after another or n at one time, is a fundamental strategy. For this goal, we define the concept of mean-frequency and develop the related frequency decomposition with the complete Szegö kernel dictionary, the latter consisting of the multiple kernels, being defined as the parameter-derivatives of the Szegö kernels. Several major energy matching pursuit type sparse representations, including greedy algorithm (GA), orthogonal greedy algorithm (OGA), adaptive Fourier decomposition (AFD), pre-orthogonal adaptive Fourier decomposition (POAFD), n-Best approximation, and unwinding Blaschke expansion, are analyzed and compared. Of which an order in reconstruction efficiency between the mentioned algorithms is given based on the detailed study of their respective remainders. The study spells out the natural connections between the multiple kernels and the related Laguerre system, and in particular, shows that both, like the Fourier series, extract out the O(n−σ) order convergence rate from the functions in the Hardy-Sobolev space of order σ>0. The existence of the n-Best approximation with the complete Szegö dictionary is proved and the related algorithm aspects are discussed. The included experiments form a significant integration part of the study, for they not only illustrate the theoretical results but also provide cross comparison between various ways of combination between the matching pursuit algorithms and the dictionaries in use. Experiments show that the complete dictionary remarkably improves approximation efficiency.

Quality controlled 2D ECG compression using adaptive fourier decomposition

Quality controlled 2D ECG compression using adaptive fourier decomposition

Nonlinear time-frequency iterative learning control for micro-robotic deposition system using adaptive Fourier decomposition approach

Nonlinear time-frequency iterative learning control for micro-robotic deposition system using adaptive Fourier decomposition approach

Simulation of non-stationary and non-Gaussian stochastic processes by the AFD-Type Sparse Representations

Simulation of a stochastic process to simultaneously meet a given marginal distribution condition and be compatible with a given covariance function is a well-known problem called the two-match problem in the sequel. We propose the adaptive Fourier decomposition (AFD) type methods as constructive steps to solve the two-match problem. AFD-Type methods used in the physical domain (e.g., time or frequency) are replacements of the Karhunen–Loève expansion, or spectral decomposition, in general, while, when being used in the probability space are replacements of the polynomial chaos or other types of chaos methods. The AFD-Type methods in both circumstances play the role of an approximation. Being compared with the Karhunen–Loève expansion, the proposed AFD-Type methods do not need to compute out the eigenvalues and eigenfunctions of the kernel integral operator defined by the target covariance function, while compared with the polynomial chaos or other chaos methods the AFD-Type methods offer a great deal of flexibility and efficiency. The proposed methods can be applied to stationary and non-stationary, weakly, and strongly non-Gaussian stochastic processes. We provide several examples to show effectiveness and efficiency of the AFD-Type methods.

Daily runoff prediction based on the adaptive fourier decomposition method and multiscale temporal convolutional network.

The non-linearity and non-stationarity of runoff series pose significant challenges to runoff forecasting, and conventional single forecasting models struggle to accurately capture the internal dynamics of the series. To address this issue, we propose a runoff prediction model named AFDM-MTCN, which combines the adaptive Fourier decomposition method (AFDM) and multiscale temporal convolutional network (MTCN). AFDM-MTCN consists of two stages: adaptive decomposition and multi-scale feature extraction. In the adaptive decomposition stage, the improved Fourier decomposition method (IFDM) is optimized using the Sparrow Search Algorithm to enhance its ability to extract temporal patterns. In the multi-scale feature extraction stage, improvements are made to the temporal convolutional network (TCN) through the use of multi-scale convolution kernels, skip connections, and depth-wise separable convolution, to capture information from multiple angles, enhance information propagation, and reduce training parameters. The model was applied to two hydrological stations in the Weihe River Basin and compared with state-of-the-art methods to assess its accuracy and feasibility. The results demonstrate that AFDM-MTCN exhibits satisfactory performance in runoff prediction. Furthermore, compared to other decomposition techniques, AFDM demonstrates stronger capability in extracting patterns from non-stationary runoff data.

Adaptive Fourier decomposition‐type sparse representations versus the Karhunen–Loève expansion for decomposing stochastic processes

The present study devotes to introducing and analyzing the use of the adaptive Fourier decomposition (AFD)‐type methods in the area of stochastic processes and random fields. We involve two types of algorithms, namely, stochastic AFD (SAFD) and stochastic pre‐orthogonal AFD (SPOAFD), for, respectively, the Hardy space format and non‐Hardy space ones, as may be regarded. We provide both their theoretical results and practical algorithms and compare them with the well adopted and, in fact, dominating Karhunen–Loève (KL)‐type expansions. The AFD methods involve a finite or infinite sequence of optimally chosen parameters; they, in contrast, do not rely on and hence not have to compute the eigenpairs of the second type Fredholm integral equation with the covariance function as the kernel. Apart from such computational conveniences, they with the same convergence rate have flexibility of choosing best suitable dictionaries for doing the specific task in the practice. We include a number of experiments showing that in terms of effectiveness, the AFD methods give better approximations before all the positive eigenvalues running out in the case the integral operator being of a finite rank or before the KL iteration step becomes excessively large: The AFD methods normally give better approximations from the very beginning of the iterations.

Algorithm of adaptive Fourier decomposition in H2(ℂ+)

In this paper, we propose an applicable algorithm of the so-called adaptive Fourier decomposition in [Formula: see text] the Hardy [Formula: see text] space on the upper half-plane [Formula: see text], which provides a new method for decomposing real-valued signals and analytic signals on the real line. As a by-product, a new approach for computing the Hilbert transform on the real line is also given. Numerical experiments are used to demonstrate the performance of the proposed algorithm.

The adaptive Fourier decomposition for financial time series

This paper explores the application of the adaptive Fourier decomposition (AFD) to a deconstruction of financial time series. After the decomposition of the time series into mono-components through AFD, we reconstruct the mono-components to obtain the series’ trend and the detailed components. AFD is compared with the Empirical Mode Decomposition (EMD) and Fourier decomposition method (FDM), which could decompose time series into trends and detailed components as well. The results based on the data from stock, commodity, exchange rate, and carbon markets show that, compared to EMD and FDM, the trends extracted by AFD are more sensitive to the peaks so that they can generally track the tendencies of the financial time series better with fewer energy differences. Besides, the detailed components of AFD better reflect the structural breaks of the original time series.

Stimulus-Stimulus Transfer Based on Time-Frequency-Joint Representation in SSVEP-Based BCIs.

Brain-computer interfaces (BCIs) based on steady-state visual evoked potential (SSVEP) require extensive and costly calibration to achieve high performance. Using transfer learning to re-use existing calibration data from old stimuli is a promising strategy, but finding commonalities in the SSVEP signals across different stimuli remains a challenge. This study presents a new perspective, namely time-frequency-joint representation, in which SSVEP signals corresponding to different stimuli can be synchronized, and thus can emphasize common components. According to this time-frequency-joint representation, an adaptive decomposition technique based on the multi-channel adaptive Fourier decomposition (MAFD) is proposed to adaptively decompose SSVEP signals of different stimuli simultaneously. Then, common components can be identified and transferred across stimuli. A simulation study on public SSVEP datasets demonstrates that the proposed stimulus-stimulus transfer method has the ability to extract and transfer these common components across stimuli. By using calibration data from eight source stimuli, the proposed stimulus-stimulus transfer method can generate SSVEP templates of other 32 target stimuli. It boosts the ITR of the stimulus-stimulus transfer based recognition method from 95.966 bits/min to 123.684 bits/min. By extracting and transfer common components across stimuli in the proposed time-frequency-joint representation, the proposed stimulus-stimulus transfer method produces good classification performance without requiring calibration data of target stimuli. This study provides a synchronization standpoint to analyze and model SSVEP signals. In addition, the proposed stimulus-stimulus method shortens the calibration time and thus improve comfort, which could facilitate real-world applications of SSVEP-based BCIs.

Weighted singular value decomposition basis of Szegő kernel and its applications to signal reconstruction and denoising

For any circular analytic signal f+(eit)∈H2(T), if the sequence {ak}k=0∞ in the unit disc D satisfies the hyperbolic non-separability condition ∑k=0∞(1−|ak|)=∞, it is known that the first N+1 term approximation function based on the Takenaka–Malmquist system (SN+1f+)(eit)=∑k=0N〈f+(eit),Bk(eit)〉Bk(eit),can approximate the analytic signal f+(eit) in the L2-norm sense, where Bk(eit)=1−|ak|21−ak¯eit∏l=0k−1eit−al1−al¯eit, k∈N is the Takenaka–Malmquist system which can be seen as a generalization of Fourier series. We find that (SN+1f+)(eit) can also be represented by the standard basis, Lagrange basis and weighted SVD basis. Different bases have different merits. The standard basis is simple. The Lagrange basis has a neat and compact form, which is very convenient for theoretical analysis. With the Takenaka–Malmquist system, the first N+1 term approximation function (SN+1f+)(eit) is able to write in an iterative form. The Takenaka–Malmquist system has the advantage of inheritance, and the point ak∈D can be easily selected step by step through the greedy principle. In this paper, we focus our attention on the weighted SVD basis. We give some orthogonal properties of the weighted SVD basis. It demonstrates that any real signal f(eit)∈L2(T) can be precisely reconstructed within the given error range by the weighted SVD basis decomposition. The experimental results reveal that the algorithm based on weighted SVD basis is more stable and accurate. Furthermore, compared with classical Fourier decomposition and adaptive Fourier decomposition algorithms, the experimental results show that the weighted SVD basis decomposition is more insensitive to noise. This will provide a new method for signal denoising.

A Precise Frequency Recognition Method of Short-time SSVEP Signals Based on Signal Extension.

Improving the Information Transfer Rate (ITR) is a popular research topic in steady-state visual evoked potential (SSVEP)-based brain-computer interfaces (BCIs). The higher recognition accuracy of short-time SSVEP signal is critical to improving ITR and achieving high-speed SSVEP-BCIs. However, the existing algorithms have unsatisfactory performance on recognizing short-time SSVEP signals, especially for calibration-free methods. This study for the first time proposed improving the recognition accuracy of short-time SSVEP signals based on the calibration-free method by extending the SSVEP signal length. A signal extension model based on Multi-channel adaptive Fourier decomposition with differ-ent Phase (DP-MAFD) is proposed to achieve signal exten-sion. Then the Canonical Correlation Analysis based on signal extension (SE-CCA) is proposed to complete the recognition and classification of SSVEP signals after exten-sion. The similarity study and SNR comparison analysis on public SSVEP datasets demonstrate that the proposed signal extension model has the ability to extend SSVEP signals. The classification results show that the pro-posed method outperforms Canonical Correlation Analysis (CCA) and Filter Bank Canonical Correlation Analysis (FBCCA) significantly in the measure of classification accu-racy and information transmission rate (ITR), especially for short-time signals. The highest ITR of SE-CCA is improved to 175.61 bits/min at around 1s, while CCA is 100.55 bits/min at 1.75s and FBCCA is 141.76 bits/min at 1.25s. The signal extension method can improve the recognition accuracy of short-time SSVEP signals and further improve the ITR of SSVEP-BCIs.

Heartbeat and Respiration Rate Prediction Using Combined Photoplethysmography and Ballisto Cardiography

Owing to the recent trends in remote health monitoring, real-time applications for measuring Heartbeat Rate and Respiration Rate (HARR) from video signals are growing rapidly. Photo Plethysmo Graphy (PPG) is a method that is operated by estimating the infinitesimal change in color of the human face, rigid motion of facial skin and head parts, etc. Ballisto Cardiography (BCG) is a nonsurgical tool for obtaining a graphical depiction of the human body’s heartbeat by inducing repetitive movements found in the heart pulses. The resilience against motion artifacts induced by luminance fluctuation and the patient’s mobility variation is the major difficulty faced while processing the real-time video signals. In this research, a video-based HARR measuring framework is proposed based on combined PPG and BCG. Here, the noise from the input video signals is removed by using an Adaptive Kalman filter (AKF). Three different algorithms are used for estimating the HARR from the noise-free input signals. Initially, the noise-free signals are subjected to Modified Adaptive Fourier Decomposition (MAFD) and then to Enhanced Hilbert vibration Decomposition (EHVD) and finally to Improved Variation mode Decomposition (IVMD) for attaining three various results of HARR. The obtained values are compared with each other and found that the EHVD is showing better results when compared with all the other methods.