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In this paper, we investigate sparse representations of approximation to identity via time–space fractional heat equations: \[ \left\{ \begin{aligned} & \partial^{\beta}_{t}u(t,x)=-\nu(-\Delta)^{\alpha/2}u(t,x),\quad (t,x)\in\mathbb R^{1+n}_{+};\cr & u(0,x)=f(x),\ x\in\mathbb R^{n}. \end{aligned}\right. \] { ∂ 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(\cdot,\cdot) $ u (⋅ , ⋅) to the above problem. This deficiency makes it difficult to verify the boundary vanishing condition of $ u(\cdot,\cdot) $ 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. [ABSTRACT FROM AUTHOR]

We first utilized the Bayes positive diagonal BEKK generalized autoregressive conditional heteroskedasticity (Bayes-pdBEKK-GARCH) model to evaluate the risk spillovers between the e-commerce and logistics, then applied the adaptive Fourier decomposition method to measure the extent of these spillovers and detect structural changes. The results showed that there were structural breaks in both markets, which may lead to extreme risks. At last, we applied the GARCH-copula quantile regression model to analyze the extreme risks. We found that: (1) there were asymmetric volatility spillovers and positive correlations between them. (2) The dynamic risk spillovers exhibited heterogeneity over time. The logistics market had a smaller downside risk spillover, while the e-commerce market had a stronger upside risk spillover. (3) The study indicated that important events, such as the Chinese stock market crash, the Sino-U.S. trade friction, the COVID-19 epidemic, and the "either-or choice" monopoly policy of e-commerce platforms, had a significant influence on them, resulting in dramatic risk spillovers. [ABSTRACT FROM AUTHOR]

We first utilized the Bayes positive diagonal BEKK generalized autoregressive conditional heteroskedasticity (Bayes-pdBEKK-GARCH) model to evaluate the risk spillovers between the e-commerce and logistics, then applied the adaptive Fourier decomposition method to measure the extent of these spillovers and detect structural changes. The results showed that there were structural breaks in both markets, which may lead to extreme risks. At last, we applied the GARCH-copula quantile regression model to analyze the extreme risks. We found that: (1) there were asymmetric volatility spillovers and positive correlations between them. (2) The dynamic risk spillovers exhibited heterogeneity over time. The logistics market had a smaller downside risk spillover, while the e-commerce market had a stronger upside risk spillover. (3) The study indicated that important events, such as the Chinese stock market crash, the Sino-U.S. trade friction, the COVID-19 epidemic, and the "either-or choice" monopoly policy of e-commerce platforms, had a significant influence on them, resulting in dramatic risk spillovers.

The paper promotes a new sparse approximation for fractional Fourier transform, which is based on adaptive Fourier decomposition in Hardy-Hilbert space on the upper half-plane. Under this methodology, the local polynomial Fourier transform characterization of Hardy space is established, which is an analog of the Paley-Wiener theorem. Meanwhile, a sparse fractional Fourier series for chirp L 2 function is proposed, which is based on adaptive Fourier decomposition in Hardy-Hilbert space on the unit disk. Besides the establishment of the theoretical foundation, the proposed approximation provides a sparse solution for a forced Schr o ¨ dinger equations with a harmonic oscillator. [ABSTRACT FROM AUTHOR]

AbstractIn 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. [ABSTRACT FROM AUTHOR]

In various applications, the adoption of optimal energy matching pursuit with dictionary elements is common. When the dictionary elements are indexed by parameters within a bounded region, exhaustion-type algorithms can be employed. This article aims to investigate a process that converts the optimal parameter selection in unbounded regions to a bounded and closed (compact) sub-domain. Such a process provides accessibility for energy matching pursuit in a wide range of applications. The paper initially focuses on the open unit disc and the upper-half complex plane, introducing adaptive Fourier decomposition as the underlying methodology. It then extends this concept to general Hilbert spaces with a dictionary and Bochner spaces for random signals. Computational examples are included to illustrate the concepts presented. [ABSTRACT FROM AUTHOR]

The shaft-rate electromagnetic field is a crucial feature of ships and undersea vehicles. However, due to the presence of static electromagnetic fields, the signal-to-noise ratio of the shaft-rate electromagnetic fields is greatly reduced. To effectively detect weak shaft-rate electromagnetic field signals under low signal-to-noise ratio conditions, this paper proposed a signal extraction method based on adaptive Fourier decomposition(AFD). It could adaptively decompose complex non-stationary signals from low frequency to high frequency into a series of single components with instantaneous frequency. Through simulation and measured data processing, the results show that this algorithm can overcome the shortcomings of methods such as short-time Fourier transform, wavelet transform, and empirical mode decomposition. It can quickly and effectively extract shaft-rate electromagnetic field signals, thereby providing a reference for positioning and tracking other ships and undersea vehicles.

This study presents a cutting-edge approach to design iterative learning control (ILC) in micro-robotic deposition systems, utilizing nonlinear time-frequency analysis through adaptive Fourier decomposition (AFD). While ILC has demonstrated its effectiveness in achieving precise trajectory tracking, achieving a balance between robustness and convergence can be challenging. To address this challenge, we introduce a novel nonlinear time-frequency ILC design from a signal processing perspective, which exploits an advanced version of Fourier decomposition called AFD. By employing adaptive basis functions, AFD enables fast energy convergence during the control process. To reduce noise amplification and system delay, we propose a phase-lead ILC algorithm with zero amplitude attenuation. Additionally, we introduce a tunable bandwidth L-Q filter to achieve an optimal trade-off between robustness and convergence. The filter's bandwidth is adaptively adjusted based on the frequency content of the system, with a narrower bandwidth for low-frequency signals to accelerate convergence and a wider bandwidth for high-frequency signals to enhance robustness. Simulation results demonstrate the exceptional performance of the proposed ILC design in a micro-robotic deposition system. [ABSTRACT FROM AUTHOR]

BackgroundThe 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.MethodOur 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.ResultThe 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

In this paper, we propose an applicable algorithm of the so-called adaptive Fourier decomposition in H 2 (ℂ +) the Hardy H 2 space on the upper half-plane ℂ + , 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. [ABSTRACT FROM AUTHOR]

In the slice Hardy space over the unit ball of quaternions, we introduce the slice hyperbolic backward shift operator S a with the decomposition process f = e a ⟨ f , e a ⟩ + B a ∗ S a f , where e a denotes the slice normalized Szegö kernel and B a the slice Blaschke factor. Iterating the above decomposition process, a corresponding maximal selection principle gives rise to the slice adaptive Fourier decomposition. This leads to a adaptive slice Takenaka–Malmquist orthonormal system. [ABSTRACT FROM AUTHOR]

Positive-instantaneous frequency representation for transient signals has always been a great concern due to its theoretical and practical importance, although the involved concept itself is paradoxical. The desire and practice of uniqueness of such frequency representation (decomposition) raise the related topics in approximation. During approximately the last two decades there has formulated a signal decomposition and reconstruction method rooted in harmonic and complex analysis giving rise to the desired signal representations. The method decomposes any signal into a few basic signals that possess positive instantaneous frequencies. The theory has profound relations to classical mathematics and can be generalized to signals defined in higher dimensional manifolds with vector and matrix values, and in particular, promotes kernel approximation for multi-variate functions. This article mainly serves as a survey. It also gives two important technical proofs of which one for a general convergence result (Theorem 3.4), and the other for necessity of multiple kernel (Lemma 3.7). Expositorily, for a given real-valued signal f one can associate it with a Hardy space function F whose real part coincides with f. Such function F has the form F = f + iH f, where H stands for the Hilbert transformation of the context. We develop fast converging expansions of F in orthogonal terms of the form where Bk's are also Hardy space functions but with the additional properties The original real-valued function f is accordingly expanded which, besides the properties of ρk and θk given above, also satisfies Real-valued functions f (t)= ρ(t)cos θ(t) that satisfy the condition are called mono-components. If f is a mono-component, then the phase derivative θ′(t) is defined to be instantaneous frequency of f. The above described positive-instantaneous frequency expansion is a generalization of the Fourier series expansion. Mono-components are crucial to understand the concept instantaneous frequency. We will present several most important mono-component function classes. Decompositions of signals into mono-components are called adaptive Fourier decompositions (AFDs). We note that some scopes of the studies on the 1D mono-components and AFDs can be extended to vector-valued or even matrix-valued signals defined on higher dimensional manifolds. We finally provide an account of related studies in pure and applied mathematics. [ABSTRACT FROM AUTHOR]

Sparse (fast) representations of deterministic signals have been well studied. Among other types, there exists one called adaptive Fourier decomposition (AFD) for functions in analytic Hardy spaces. Through the Hardy space decomposition of the L2‐space, the AFD algorithm also gives rise to sparse representations of signals of finite energy. To deal with multivariate signals, the general Hilbert space context comes into play. The multivariate counterpart of AFD in general Hilbert spaces with a dictionary has been named preorthogonal AFD (POAFD). In the present study, we generalize AFD and POAFD to random analytic signals through formulating stochastic analytic Hardy spaces and stochastic Hilbert spaces. To analyze random analytic signals, we work on two models, both being called stochastic AFD, or SAFD in brief. The two models are respectively made for (i) those expressible as the sum of a deterministic signal and an error term (SAFDI) and for (ii) those from different sources obeying certain distributive law (SAFDII). In the later part of the paper, we drop off the analyticity assumption and generalize the SAFDI and SAFDII to what we call stochastic Hilbert spaces with a dictionary. The generalized methods are named as stochastic POAFDs, SPOAFDI and SPOAFDII. Like AFDs and POAFDs for deterministic signals, the developed stochastic POAFD algorithms offer powerful tools to approximate and thus to analyze random signals. [ABSTRACT FROM AUTHOR]

Evolved from the conventional Fourier decomposition based on a pre-defined basis, Adaptive Fourier decomposition (AFD) uses adaptive basis to achieve the fast energy convergence. This paper extends the AFD to the multi-channel case, which finds common adaptive basis across all channels. The proposed multi-channel AFD (MAFD) scheme includes the multi-channel core AFD for general signals and the multi-channel unwinding AFD for specific signals that have common inner functions. Owing to the merits of the original AFD, the MAFD can provide sparse joint time-frequency distribution by computing the transient time frequency distribution (TTFD) across channels. Simulations on synthetic and real-world signals demonstrate that the proposed scheme can find and apply the common adaptive basis with desired properties maintained by the AFD, showing high potentials in real-world applications. [ABSTRACT FROM AUTHOR]

This paper studies the carbon price fluctuation in China through the adaptive Fourier decomposition (AFD). Apart from the transient time-frequency distribution of the original AFD model, we also reconstruct the mono-components of this model to obtain the components in different time-frequency scales. Our empirical results based on the carbon price in Hubei Province demonstrate that there are three periods when the price fluctuates dramatically, mainly affected by the governmental policies about carbon emission and the development of clean energies, as well as the outbreak of COVID-19. Furthermore, the fluctuations of the price in the three identified periods are reflected in different scales. The comparison of the decomposition results and those of EMD and VMD shows that the AFD performs best in absorbing the price's useful information extracted through all these methods. [ABSTRACT FROM AUTHOR]

In view of recent developments of the study of reproducing kernel Hilbert spaces, in particular with the context the Hardy spaces on tubes, aspects of rational approximation for functions of finite energy in several complex and several real variables are developed. [ABSTRACT FROM AUTHOR]

Möbius transforms, Blaschke products and starlike functions as typical conformal mappings of one complex variable give rise to nonlinear phases with non-negative phase derivatives with the latter being defined by instantaneous frequencies of signals they represent. The positive analytic phase derivative has been a widely interested subject among signal analysts (see Gabor (1946)). Research results of the positive analytic frequency and applications appears in the literature since the middle of the 20th century. Of the positive frequency study a directly related topic is positive frequency decomposition of signals. The mainly focused methods of such decompositions include the maximal selection method and the Blaschke product unwinding method, and joint use of the mentioned methods. In this paper, we propose a class of iterative greedy algorithms based on the Blaschke product and adaptive Fourier decomposition. It generalizes the Blaschke product unwinding method by subtracting constants other than the averages of the remaining functions, aiming at larger winding numbers, and subtracting n-Blaschke forms of the remaining functions, aiming at generating larger numbers of zero-crossings, to fast reduce energy of the remaining terms. Furthermore, we give a comprehensive and rigorous proof of the converging rate in terms of the zeros of the remainders. Finite Blaschke product methods are proposed to avoid the infinite phase derivative dilemma, and to avoid the computational difficulties. [ABSTRACT FROM AUTHOR]

Few studies have considered the information in the frequency domain to detect the structural breaks in financial markets. This paper provides a mixed model that integrates BEMD, AFD and Chow test to study the fluctuation characteristics of the soybean futures price. Due to the application of AFD, the mixed model can detect the structural breaks of the price fluctuation through obtaining the high-resolution information in the frequency domain. According to our results, in general, the soybean futures price in China is mainly determined by IMF11, followed by IMF9, IMF8 and IMF7, and there exist many structural breaks of different IMFs. Interestingly, national macro-controls have opposite effects at IMF8 and IMF11. The results show the effectiveness of the proposed mixed model for detecting structural breaks of financial markets in frequency-domain by revealing the impact of external events on the soybean futures price at multi-scales. [ABSTRACT FROM AUTHOR]

This paper proposes a novel feature representation approach for heartbeat classification using single-lead electrocardiogram (ECG) signals based on adaptive Fourier decomposition (AFD). AFD is a recently developed signal processing tool that provides useful morphological features, which are referred as AFD-derived instantaneous frequency (IF) features and differ from those provided by traditional tools. The AFD-derived IF features, together with ECG landmark features and RR interval features, are trained by a support vector machine to perform the classification. The proposed method improves the average accuracy of the feature extraction-based methods, reaching a level comparable to deep learning but with less training data, and at the same time being interpretable for the learned features. It also greatly reduces the dimension of the feature set, which is a disadvantage of the feature extraction-based methods, especially for ECG signals. To evaluate the performance, the Association for the Advancement of Medical Instrumentation standard is applied to publicly available benchmark databases, including the MIT-BIH arrhythmia and MIT-BIH supraventricular arrhythmia databases, to classify heartbeats from the single-lead ECG. The overall performance is compared to selected state-of-the-art automatic heartbeat classification algorithms, including one-lead and even several two-lead-based methods. The proposed approach achieves superior balanced performance and real-time implementation. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Guangdong University of Technology is the property of Journal of Guangdong University of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Background: This study provides a detailed analysis of the daily fluctuations in coronavirus disease 2019 (COVID-19) case numbers in London from January 31, 2020 to February 24, 2022. The primary objective was to enhance understanding of the interactions among government pandemic responses, viral mutations, and the subsequent changes in COVID-19 case incidences., Methods: We employed the adaptive Fourier decomposition (AFD) method to analyze diurnal changes and further segmented the AFD into novel multi-component groups consisting of one to three elements. These restructured components were rigorously evaluated using Pearson correlation, and their effectiveness was compared with other signal analysis techniques. This study introduced a novel approach to differentiate individual components across various time-frequency scales using basis decomposition methods., Results: Analysis of London's daily COVID-19 data using AFD revealed a strong correlation between the "stay at home" directive and high-frequency components during the first epidemic wave. This indicates the need for sustained implementation of vaccination policies to maintain their effectiveness., Discussion: The AFD component method provides a comprehensive analysis of the immediate and prolonged impact of governmental policies on the spread of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). This robust tool has proven invaluable for analyzing COVID-19 pandemic data, offering critical insights that guide the formulation of future preventive and public health strategies., (Copyright and License information: Editorial Office of CCDCW, Chinese Center for Disease Control and Prevention 2024.)

Abstract The adaptive Fourier decomposition (AFD) uses an adaptive basis instead of a fixed basis in the rational analytic function and thus achieves a fast energy convergence rate. At each decomposition level, an important step is to determine a new basis element from a dictionary to maximize the extracted energy. The existing basis searching method, however, is only the exhaustive searching method that is rather inefficient. This paper proposes four methods to accelerate the AFD algorithm based on four typical optimization techniques including the unscented Kalman filter (UKF) method, the Nelder-Mead (NM) algorithm, the genetic algorithm (GA), and the particle swarm optimization (PSO) algorithm. In the simulation of decomposing four representative signals and real ECG signals, compared with the existing exhaustive search method, the proposed schemes can achieve much higher computation speed with a fast energy convergence, that is, in particular, to make the AFD possible for real-time applications.

The existing frequency-domain-based iterative learning control (ILC) methods are highly dependent on the mathematical models of the controlled systems. For linear discrete-time single-input single-output (SISO) systems with unknown mathematical models, this paper tries to present fully data-driven ILC designs in frequency domain. With the help of support vector machine (SVM), the input-output data of the linear discrete-time SISO system at the first repetition is utilised to constitute an adaptive Fourier decomposition (AFD) model. Then, based on the AFD model, a P-type ILC law and an extended D-type ILC law with data-driven determining techniques for learning gains are presented. It is noted that comparing with the conventional D-type ILC law, the newly proposed extended D-type ILC law exhibits superior tracking characteristic due to involving the frequency information during the ILC process. A numerical example is utilised to illustrate the effectiveness of the proposed ILC algorithms with the data-driven determining techniques for learning gains. [ABSTRACT FROM AUTHOR]

The emerging complex circumstances caused by economy, technology, and government policy and the requirement of low-carbon development of power grid lead to many challenges in the power system coordination and operation. However, the real-time scheduling of electricity generation needs accurate modeling of electricity demand forecasting for a range of lead times. In order to better capture the nonlinear and non-stationary characteristics and the seasonal cycles of future electricity demand data, a new concept of the integrated model is developed and successfully applied to research the forecast of electricity demand in this paper. The proposed model combines adaptive Fourier decomposition method, a new signal preprocessing technology, for extracting useful element from the original electricity demand series through filtering the noise factors. Considering the seasonal term existing in the decomposed series, it should be eliminated through the seasonal adjustment method, in which the seasonal indexes are calculated and should multiply the forecasts back to restore the final forecast. Besides, a newly proposed moth-flame optimization algorithm is used to ensure the suitable parameters of the least square support vector machine which can generate the forecasts. Finally, the case studies of Australia demonstrated the efficacy and feasibility of the proposed integrated model. Simultaneously, it can provide a better concept of modeling for electricity demand prediction over different forecasting horizons. [ABSTRACT FROM AUTHOR]

Precise heart rate (HR) estimation of individuals during physical exercise from PPG sensors using direct methods often produces erroneous output due to the occurrence of exceedingly high motion artifacts (MAs). The effect of MAs can be neutralized using non-direct algorithms such as optimization techniques using some a priori about MA. In this paper, we propose a dual-optimization technique using adaptive Fourier decomposition in conjunction with sparse spectral reconstruction (AFODSS) using a multiple measurement vectors (MMVs) model and a regularized bi-conjugate gradient focal underdetermined system solver (BCG-FOCUSS) algorithm. Using AFODSS, the average absolute error (AAE) of HR obtained on the dataset recorded from 23 subjects during their intense exercise is 1.35 beats/min. The BCG-FOCUSS algorithm used in this paper helps to speed up the convergence rate and reduce time complexity when compared with other FOCUSS algorithms. [ABSTRACT FROM AUTHOR]

This paper aims at proposing a novel 2D non-linear phase decomposition of images, which performs the image processing tasks better than the traditional Fourier transformation (linear phase decomposition), but further, it has additional mathematical properties allowing more effective image analysis, including adaptive decomposition components and positive instantaneous phase derivatives. 1D unwinding Blaschke decomposition has recently been proposed and studied. Through factorization it expresses arbitrary 1D signal into an infinite linear combination of Blaschke products. It offers fast converging positive frequency decomposition in the form of rational approximation. However, in the multi-dimensional cases, the usual factorization mechanism does not work. As a consequence, there is no genuine unwinding decomposition for multi-dimensions. In this paper, a 2D partial unwinding decomposition based on algebraic transforms reducing multi-dimensions to the 1D case is proposed and analyzed. The result shows that the fast convergence offers efficient image reconstruction. The tensor type decomposing terms are mutually orthogonal, giving rise to 2D positive frequency decomposition. The comparison results show that the proposed method outperforms the standard greedy algorithm and the most commonly used methods in the Fourier category. An application in watermarking is presented to demonstrate its potential in applications. [ABSTRACT FROM AUTHOR]

This study aims at rational approximation of a class of weighted Hardy spaces, including the classical Bergman space, the weighted Bergman spaces, the Hardy space, the Dirichlet space and the Hardy–Sobolev spaces. We will mainly concentrate in the Bergman cases in the unit disc context. The methodology of the approximation is a pre-orthogonal method, called Pre-Adaptive Fourier Decomposition. The new idea is that a function is not expanded into a basis but an orthonormal system adapted to the given function. In such way by using a unified method we obtain efficient approximations in our sequence of spaces while avoiding discussions on basis and uniqueness sets, etc. The type of function decompositions is related to direct sum decompositions of the underlying spaces into the closure of the span of a sequence of repeating reproducing kernels and the corresponding zero-based invariant subspaces that arises deep studies. [ABSTRACT FROM AUTHOR]

In this paper, we study decomposition of functions in Hardy spaces Hp(T)(1

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. [ABSTRACT FROM AUTHOR]

Background: 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., Method: 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., Result: 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., Conclusion: 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., Competing Interests: The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest., (Copyright © 2023 Lu, Yang, Qu, Qian, Liu, He, Lin and Hon.)

In this paper we consider rational approximation of functions in the Hardy spaces on the strip with . Observing that can be regarded as the direct sum of and , we give three kinds of adaptive rational approximation in enhancing the corresponding approximations in and where and We also obtain a type of approximation for subspaces in by making use of Riesz bases in these subspaces. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Signal Processing is the property of Journal of Signal Processing and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

For any circular analytic signal f + (e i t) ∈ H 2 (T) , if the sequence { a k } k = 0 ∞ in the unit disc D satisfies the hyperbolic non-separability condition ∑ k = 0 ∞ (1 − | a k |) = ∞ , it is known that the first N + 1 term approximation function based on the Takenaka–Malmquist system (S N + 1 f +) (e i t) = ∑ k = 0 N 〈 f + (e i t) , B k (e i t) 〉 B k (e i t) , can approximate the analytic signal f + (e i t) in the L 2 -norm sense, where B k (e i t) = 1 − | a k | 2 1 − a k ¯ e i t ∏ l = 0 k − 1 e i t − a l 1 − a l ¯ e i t , k ∈ N is the Takenaka–Malmquist system which can be seen as a generalization of Fourier series. We find that (S N + 1 f +) (e i t) 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 (S N + 1 f +) (e i t) is able to write in an iterative form. The Takenaka–Malmquist system has the advantage of inheritance, and the point a k ∈ 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 (e i t) ∈ L 2 (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. [ABSTRACT FROM AUTHOR]

The Dirac-type time-frequency distribution (TFD), regarded as ideal TFD, has long been desired. It, until the present time, cannot be implemented, due to the fact that there has been no appropriate representation of signals leading to such TFD. Instead, people have been developing other types of TFD, including the Wigner and the windowed Fourier transform types. This paper promotes a practical passage leading to a Dirac-type TFD. Based on the proposed function decomposition method, viz., adaptive Fourier decomposition, we establish a rigorous and practical Dirac-type TFD theory. We do follow the route of analytic signal representation of signals founded and developed by Garbo, Ville, Cohen, Boashash, Picinbono, and others. The difference, however, is that our treatment is theoretically throughout and rigorous. To well illustrate the new theory and the related TFD, we include several examples and experiments of which some are in comparison with the most commonly used TFDs. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In the literature adaptive Fourier decomposition is abbreviated as AFD that addresses adaptive rational approximation, or alternatively adaptive Takenaka–Malmquist system approximation. The AFD type approximations may be characterized as adaptive approximations by linear combinations of parameterized Szegö and higher order Szegö kernels. This note proposes two kinds of such analytic approximations of which one is called maximal-energy AFDs, including core AFD, Unwending AFD and Cyclic AFD; and the other is again linear combinations of Szegö kernels but generated through SVM methods. The proposed methods are based on the fact that the imaginary part of an analytic signal is the Hilbert transform of its real part. As consequence, when a sequence of rational analytic functions approximates an analytic signal, then the real parts and imaginary parts of the functions in the sequence approximate, respectively, the original real-valued signals and its Hilbert transform. The two approximations have the same errors in the energy sense due to the fact that Hilbert transformation is a unitary operator in the L 2 space. This paper for the first time promotes the complex analytic method for computing Hilbert transforms. Experiments show that such computational methods are as effective as the commonly used one based on FFT. [ABSTRACT FROM AUTHOR]