广东工业大学学报 ›› 2021, Vol. 38 ›› Issue (03): 65-71.doi: 10.12052/gdutxb.200094

• • 上一篇    下一篇

自适应Fourier分解思想在再生核W2 1 [a,b]空间的应用

蒋文超, 谭立辉   

  1. 广东工业大学 应用数学学院，广东 广州 510520
  • 收稿日期:2020-07-30 出版日期:2021-05-10 发布日期:2021-03-12
  • 作者简介:蒋文超(1995-)，女，硕士研究生，主要研究方向为自适应傅里叶分解在再生核空间的应用
  • 基金资助:
    广东省自然科学杰出青年基金资助项目(Yq2014060)

Application of the Principle of Adaptive Fourier Decomposition in Reproducing Kernel W2 1[a,b] Space

Jiang Wen-chao, Tan Li-hui   

  1. School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510520, China
  • Received:2020-07-30 Online:2021-05-10 Published:2021-03-12

HTML

PDF (PC)

2569

摘要： 在再生核$W_2^1[a, b]$空间中研究自适应正交贪婪分解算法, 利用能量下降最快的原理自适应性地构造出最佳$n$项逼近函数, 并从理论上证明其收敛性成立。最后, 实验验证了在$W_2^1[a, b]$再生核空间中, 利用正交贪婪原理构造的$n$项数值原函数比用等分结点构造出的最佳$n$项数值原函数收敛效果更优。

关键词: 最佳数值原函数, 正交贪婪分解算法, 自适应Fourier分解, 数值逼近

Abstract: Te adaptive orthogonal greedy decomposition algorithm in the reproducing kernel $W_2^1[a, b]$-space is studied. The optimal n-term approximation function is adaptively constructed based on the principle of the fastest energy descent, and the convergence of this algorithm is proved theoretically. Finally, an experiment is used to verify that in the reproducing kernel $W_2^1[a, b]$-space, the best n-term numerical original function constructed by the orthogonal greedy principle has a better convergence effect than the best n-term numerical original function constructed with equal division nodes.

Key words: optimal numerical primitive function, orthogonal greedy algorithm, adaptive Fourier decomposition, numerical approximation

中图分类号: 

  • O242.2
[1] 石斌, 郭俊锋. 基于过完备字典稀疏表示振动信号压缩感知方法[J]. 机械设计与制造工程, 2018, 47(5): 55-59.
SHI B, GUO J F. Compressed sensing method based on over-complete dictionary and sparse representation of vibration signal [J]. Mechanical Design and Manufacturing Engineering, 2018, 47(5): 55-59.
[2] ROSENTHAL A, RAZANSKY D, NTZIACH- RISTOS V. Quantitative optoacoustic signal extraction using sparse signal representation [J]. IEEE Transactions on Medical Imaging, 2009, 28(12): 1997-2006.
[3] PROTTER M, YAVNEH I, ELAD M. Closed- form MMSE estimation for signal denoising under sparse representation modeling over a unitary dictionary [J]. IEEE Transactions on Signal Processing, 2010, 58(7): 3471-3484.
[4] DAUBECHIES I. Ten lectures on wavelets [J]. Computers in Physics, 1992, 93(3): 1671-1671.
[5] COHEN L. Time-frequency analysis[M]. New Jersey: Prentice Hall, 1995.
[6] HUANG N E, SHEN Z, LONG S R, et al. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis [J]. Proceedings Mathematical Physical & Engineering Sciences, 1998, 454(1971): 903-995.
[7] GABOR D. Theory of communication [J]. Journal of Institute of Electrical Engineers, 1946, 93(26): 429-441.
[8] GEORGOULAS G, TSOUMAS I, ANTONINO DAVIU J, et al. Automatic pattern identification based on the complex empirical mode decomposition of the startup current for the diagnosis of rotor asymmetries in asynchronous machines [J]. IEEE Transactions on Industrial Electronics, 2014, 61(9): 4937-4946.
[9] YEH J, FAN S, SHIEH J, et al. Human heart beat analysis using a modified algorithm of detrended fluctuation analysis based on empirical mode decomposition [J]. Medical Engineering and Physics, 2009, 31(1): 92-100.
[10] LEI Y G, LIN J, HE Z J, et al. A review on empirical mode decomposition in fault diagnosis of rotating machinery [J]. Mechanical System and Signal Processing, 2013, 35(1): 108-126.
[11] HUANG N E, DAUBECHIES I, HOU T Y. Adaptive data analysis: theory and applications [J]. Philosophical Transactions of the Royal Society A-Mathematical Physical and Engineering Sciences, 2016, 374: 20-65.
[12] SHARPLEY R C, VATCHEV V. Analysis of the intrinsic mode functions [J]. Constructive Approximation, 2006, 24(1): 17-47.
[13] DOROSLOVACKI M I. On nontrivial analytic signals with positive instantaneous frequency [J]. Signal Processing, 2003, 83(3): 655-658.
[14] QIAN T, CHEN Q H, Li L Q. Analytic unit quadrature signals with nonlinear phase [J]. Physica D: Nonlinear Phenomena, 2005, 203: 80-87.
[15] TAN L H, YANG L H, HUANG D R. The structure of instantaneous frequencies of periodic analytic signals [J]. Science China, 2010, 53(2): 347-355.
[16] QIAN T, TAN L H. Characterizations of mono- components: the blaschke and starlike types [J]. Complex Analysis and Operator Theory, 2018, 12: 1383-1399.
[17] BULTHEEL A. Orthogonal rational functions[M]. Cambridge: Cambridge University Press, 1999: 123-126.
[18] NINNESS B, GUSTAFSSON F. A unifying construction of orthonormal bases for system identification [J]. IEEE Transactions on Automatic Control, 2002, 42(4): 515-521.
[19] HUSEYIN A, NINNESS B. Orthonormal basis functions for modelling continuous time systems [J]. Signal Processing, 1999, 77(3): 261-274.
[20] QIAN T, SPROBIG W, WANG Y B. Adaptive fourier decomposition of functions in quaternionic hardy spaces [J]. Mathematical Methods in the Applied Sciences, 2012, 35(1): 43-64.
[21] SHIM B, WANG J, KWON S. Generalized orthogonal matching pursuit [J]. IEEE Transaction on Signal Processing, 2012, 60(4): 6202-6216.
[22] QIAN T, WANG Y B. Adaptive fourier series-a variation of greedy algorithm[M]. New York: Springer-Verlag, 2011: 279-293.
[23] ZHANG L M. Adaptive fourier decomposition based time-frequency analysis [J]. Journal of Electronic Science and Technology, 2014, 12(2): 201-205.
[24] TAN C Y, ZHANG L M, WU H T. A novel blaschke unwinding adaptive fourier decomposition based signal compression algorithm with application on ECG signals [J]. IEEE Journal of Biomedical and Health Informatics, 2019, 23(2): 672-682.
[25] MI W, QIAN T. Frequency-domain identification: an algorithm based on an adaptive rational orthogonal system [J]. Automatica, 2012, 48(6): 1154-1162.
[26] 钱涛. 自适应Fourier变换: 一个贯穿复几何, 调和分析及信号分析的数学方法[M]. 北京: 科学出版社, 2015: 25-33.
[27] 崔明根, 邓中兴. W21空间中的最佳插值逼近算子[J]. 计算数学, 1986, 8(2): 209-216.
CUI M G, DENG Z X. The best interpolation approximation operator in W21-space [J]. Computational Mathematics, 1986, 8(2): 209-216.
[28] 崔明根, 吴勃英. 再生核空间的数值分析[M]. 北京: 科学出版社, 2004: 124-128.
[29] 吴勃英, 林迎珍. 应用型再生核空间[M]. 北京: 科学出版社, 2012: 130-141.
[1] 谢伟翔, 莫艳. 基于复值稀疏贝叶斯的Hilbert变换计算方法[J]. 广东工业大学学报, 2021, 38(05): 48-51.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 2017 《广东工业大学学报》编辑部
地址：广州市东风东路729号广东工业大学 邮编：510090
电话：020-37626653,020-37626139,020-37626396
邮箱：xbzrb@gdut.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发