Continuous Data Assimilation Using General Interpolant Observables

We present a new continuous data assimilation algorithm based on ideas that have been developed for designing finite-dimensional feedback controls for dissipative dynamical systems, in particular, in the context of the incompressible two-dimensional Navier–Stokes equations. These ideas are motivated by the fact that dissipative dynamical systems possess finite numbers of determining parameters (degrees of freedom) such as modes, nodes and local spatial averages which govern their long-term behavior. Therefore, our algorithm allows the use of any type of measurement data for which a general type of approximation interpolation operator exists. Under the assumption that the observational measurements are free of noise, our main result provides conditions, on the finite-dimensional spatial resolution of the collected data, sufficient to guarantee that the approximating solution, obtained by our algorithm from the measurement data, converges to the unknown reference solution over time. Our algorithm is also applicable in the context of signal synchronization in which one can recover, asymptotically in time, the solution (signal) of the underlying dissipative system that is corresponding to a continuously transmitted partial data.     

On a convergence of the rational-trigonometric-polynomial approximations realized by the roots of the Laguerre polynomials

The paper considers a problem of approximation of functions by means of their finite number of Fourier coefficients. Convergence acceleration of approximations by the truncated Fourier series is achieved by application of polynomial and rational correction functions. Rational corrections include unknown parameters whose determination is a crucial problem. We consider an approach connected with the roots of the Laguerre polynomials and study the rates of convergence of such approximations.     

On the spectrum of periodic signals

In theory, there are many methods for the representation of signals. In practice, however, Fourier analysis involving the resolution of signals into sinusoidal components is used widely. There are several methods for Fourier analysis available for representation of signals. If the signal is periodic, then the Fourier series is used to represent the signal in terms of a set of harmonically related sinewaves. In this article new formulae for evaluating the trigonometric Fourier series coefficients when the signal under consideration is polynomial are developed by changing the integration to a derivation form. The solution presented here is an extension of the formulae proposed by Al-Smadi and Wilkes to the trigonometric Fourier series. The proposed technique is a powerful tool that can be used for solving practical science and engineering problems without excessive tedium.     

Data-driven optimal tracking control of switched linear systems

This paper addresses the optimal tracking control for switched linear systems with unknown dynamics. We convert the problem into an optimal control problem of the augmented switched systems. In view of the augmented systems, we propose a data-driven switched linear quadratic regular algorithm for obtaining the optimal switching signal under unknown system dynamics. It is proved that the optimal switching signal will not cause Zeno behavior and can make the system stable. Besides, with the proposed algorithm, we just need to identify an autonomous system instead of the original systems, which has fewer parameters to be determined. A numerical example is given to illustrate the validity of the main results.     

Fast computation of spectral centroids

The spectral centroid of a signal is the curve whose value at any given time is the centroid of the corresponding constant-time cross section of the signal’s spectrogram. A spectral centroid provides a noise-robust estimate of how the dominant frequency of a signal changes over time. As such, spectral centroids are an increasingly popular tool in several signal processing applications, such as speech processing. We provide a new, fast and accurate algorithm for the real-time computation of the spectral centroid of a discrete-time signal. In particular, by exploiting discrete Fourier transforms, we show how one can compute the spectral centroid of a signal without ever needing to explicitly compute the signal’s spectrogram. We then apply spectral centroids to an emerging biometrics problem: to determine a person’s heart and breath rates by measuring the Doppler shifts their body movements induce in a continuous wave radar signal. We apply our algorithm to real-world radar data, obtaining heart- and breath-rate estimates that compare well against ground truth.     

The arithmetic fourier transform and the visual pathway

The Arithmetic Fourier Transform is an algorithm for the computation of Fourier coefficients. In this paper it is extended to the computation of double Fourier coefficients by a repeated sum. We discuss the relevance of the repeated sum algorithm to signal processing by neurons in the visual pathway.     

Chaos control on autonomous and non-autonomous systems with various types of genetic algorithm-optimized weak perturbations

Recently, we proposed a chaos control strategy with weak Fourier signals optimized by using a genetic algorithm (GA) and demonstrated its merits in controlling Lorenz and Rössler systems (Physical Review E, 2004). In this continuation work, performance of various types of signals, namely periodic continuous, periodic discrete, and constant bias (non-periodic), applied to an autonomous (Rössler) system and a non-autonomous (Murali–Lakshmanan–Chua, MLC) system are investigated. An index of relative robustness is proposed for measuring the noise-resisting ability of the control signals. The results reveal that the constant signal has the strongest noise-resisting ability, the periodic pulse signal has the weakest, and the Fourier signal falls in between. Phase modulation generally shortens the transient time period and is additionally beneficial to non-autonomous systems in minimizing significantly the signal power. By searching with the present GA-optimization, it is demonstrated that the minimum-power signal for controlling the non-autonomous (MLC) system is the signal with a frequency exactly the same as that of the system forcing but with phase modulation. The effectiveness of the GA-optimized signals of extremely low power employed in alternatively switching control of non-autonomous systems is also demonstrated.     

A discrete Fourier transform based on Simpson's rule

Fourier analysis plays a vital role in the analysis of continuous‐time signals. In many cases, we are forced to approximate the Fourier coefficients based on a sampling of the time signal. Hence, the need for a discrete transformation into the frequency domain giving rise to the classical discrete Fourier transform. In this paper, we present a transformation that arises naturally if one approximates the Fourier coefficients of a continuous‐time signal numerically using the Simpson quadrature rule. This results in a decomposition of the discrete signal into two sequences of equal length. We show that the periodic discrete time signal can be reconstructed completely from its discrete spectrum using an inverse transform. We also present many properties satisfied by this transform. Copyright © 2012 John Wiley & Sons, Ltd.     

经济时间序列的连续参数小波网络预测模型

本文利用连续小波变换方法,给出一种连续参数小波网络。网络参数的学习采用一种类似神经网络的后向传播学习算法的随机梯度算法。另外,提出了一种借助小波分析理论指导网络参数赋初值的方法。进一步,通过对中国进出口贸易额时间序列预测建模的研究和仿真预测,提出了用连续参数小波网络建立经济时间序列预测模型的一般步骤和方法。预测结果表明,此模型具有较好的泛化、学习能力,可以有效地在数值上逼近时间序列难以定量描述的相互关系,所以利用连续参数小波网络建立的时间序列预测模型有较高的预测精度。     

窗口Fourier变换反演公式的级数表示——献给陆善镇教授75华诞

本文研究连续窗口Fourier变换的反演公式.与经典的积分重构公式不同,本文证明当窗函数满足合适的条件时,窗口Fourier变换的反演公式可以表示为一个离散级数.此外,本文还研究这一重构级数的逐点收敛及其在Lebesgue空间的收敛性.对于L^2空间,本文给出重构级数收敛的充分必要条件.     

An automatic augmented galerkin method for singular integral equations with Hilbert kernel

In [1, 2], described a Chebyshev series method for the numerical solution of integral equations with three automatic algorithms for computing two regularization parameters,C _f andr. Here we describe a Fourier series expansion method for a class of singular integral equations with Hilbert kernel and constant coefficients with using a new automatic algorithm.     

基于离散观测下Cauchy-OU过程的最大似然估计

基于离散观测样本,本文研究Cauchy-OU过程的参数估计问题.在大多数情况下,离散时间的最大似然函数是不能直接计算出来的,因此采用傅里叶变换及Gaver-Stehfest算法,构造似然函数的一个显式逼近序列,且该序列收敛于真实(但未知)的似然函数.最后,采用最大似然估计法估计出未知参数.仿真实验表明,所得到的参数估计是比较准确且稳定的.     

Parameter estimation of partially observed continuous time stochastic processes via the EM algorithm

An algorithm is presented for the problem of maximum likelihood (ML) estimation of parameters of partially observed continuous time random processes. This algorithm is an extension of the EM algorithm [3] used in the time series literature, and preserves its main features. It is then applied to the problem of parameter estimation of continuous time, finite state or infinite state (diffusions) Markov processes observed via a noisy sensor. The algorithm in general involves iterations of non-linear smoothing with known parameters and then a non-stochastic maximization. For special cases, including linear models and AR/ARMA processes observed in white noise, each iteration is easily performed with finite dimensional filters. Finally, the algorithm is applied to parameter estimation of “randomly slowly varying” linear systems observed in white noise, and explicit results are derived.     

Optimum signal and image recovery by the method of alternating projections in fractional Fourier domains

This paper presents a signal and image recovery scheme by the method of alternating projections onto convex sets in optimum fractional Fourier domains. It is shown that the fractional Fourier domain order with minimum bandwidth is the optimum fractional Fourier domain for the method employing alternating projections in signal recovery problems. Following the estimation of optimum fractional Fourier transform orders, incomplete signal is projected onto different convex sets consecutively to restore the missing part. Using a priori information in optimum fractional Fourier domains, superior results are obtained compared to the conventional Fourier domain restoration. The algorithm is tested on 1-D linear frequency modulated signals, real biological data and 2-D signals presenting chirp-type characteristics. Better results are obtained in the matched fractional Fourier domain, compared to not only the conventional Fourier domain restoration, but also other fractional Fourier domains.     

Static and dynamic stability for geometrically nonlinear governing equations of elastic thin shallow shells

In this study, the governing equations for large deflection of elastic thin shallow shells are deduced into an algebraic cubic equation to determine the unknown coefficient of the assumed deflection by applying Galerkin's method in combination with the algebraic polynomial and Fourier series. For the dynamic problem, the coefficient is replaced by an unknown function of time; after the same process is applied, the governing equations are deduced to be a nonlinear ODE of order two called the Duffing equation, and its analytical solution is known. The combination of the algebraic polynomial and Fourier series gives very rapid convergence in the asymptotic solutions.     

Complete signal processing bases and the Jacobi group

The continuous windowed Fourier and wavelet transforms are created from the actions of the Heisenberg and affine groups, respectively. Both wavelet and windowed Fourier bases are known to be complete; that is, the only signal which is orthogonal to every element of each basis is the zero signal. The Jacobi group is a group which contains both the Heisenberg and affine groups, and it can also be used to produce bases for signal processing. This paper investigates completeness for bases of one and two real variables which are produced by the Jacobi group.     

Band-limited signal extrapolation by truncated Bernstein polynomials

An algorithm is given for everywhere extrapolating a band-limited signal known only on an interval of arbitrary finite length. The scheme utilizes a finite number of equally spaced samples of the given function and provides a time-limited polynomial approximation. The approximation functions are shown to converge everywhere pointwise and uniformly in any compact interval to the band-limited signal. When the original band-limited signal is also Lebesgue integrable it is also established that the Fourier transform of the approximating signal converges uniformly to the Fourier transform of the original signal.     

A fast alternating minimization algorithm for total variation deblurring without boundary artifacts

Recently, a fast alternating minimization algorithm for total variation image deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang (2008) [32]. The method in a nutshell consists of a discrete Fourier transform-based alternating minimization algorithm with periodic boundary conditions and in which two fast Fourier transforms (FFTs) are required per iteration. In this paper, we propose an alternating minimization algorithm for the continuous version of the total variation image deblurring problem. We establish convergence of the proposed continuous alternating minimization algorithm. The continuous setting is very useful to have a unifying representation of the algorithm, independently of the discrete approximation of the deconvolution problem, in particular concerning the strategies for dealing with boundary artifacts. Indeed, an accurate restoration of blurred and noisy images requires a proper treatment of the boundary. A discrete version of our continuous alternating minimization algorithm is obtained following two different strategies: the imposition of appropriate boundary conditions and the enlargement of the domain. The first one is computationally useful in the case of a symmetric blur, while the second one can be efficiently applied for a nonsymmetric blur. Numerical tests show that our algorithm generates higher quality images in comparable running times with respect to the Fast Total Variation deconvolution algorithm.     

Consecutive minimum phase expansion of physically realizable signals with applications

In digital signal processing, it is a well know fact that a causal signal of finite energy is front loaded if and only if the corresponding analytic signal, or the physically realizable signal, is a minimum phase signal, or an outer function in the complex analysis terminology. Based on this fact, a series expansion method, called unwinding adaptive Fourier decomposition (AFD), to give rise to positive frequency representations with rapid convergence was proposed several years ago. It appears to be a promising positive frequency representation with great potential of applications. The corresponding algorithm, however, is complicated due to consecutive extractions of outer functions involving computation of Hilbert transforms. This paper is to propose a practical algorithm for unwinding AFD that does not depend on computation of Hilbert transform, but, instead, factorizes out the Blaschke product type of inner functions. The proposed method significantly improves applicability of unwinding AFD. As an application, we give the associated Dirac‐type time‐frequency distribution of physically realizable signals. Copyright © 2015 John Wiley & Sons, Ltd.     

Calculation of the field of a source situated inside a thin sphere with a crack

A rigorous solution is obtained of the problem of diffraction by a sphere with a crack in terms of a system of linear algebraic equations of second kind which converges well for the Fourier coefficients of the unknown field. The matrix operator of the system has completely continuous form which guarantees the convergence of the method of reduction. For certain values of the parameters the operator has small norm. The solution is convenient for computer calculations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 89, pp. 270–274, 1979.