Please wait a minute...
北京大学期刊网 | 作者  审稿人  工作人员

首页   |   关于   |   浏览   |   投稿指南   |   新闻公告
数学进展 - 2018, Vol. 47(3): 321-347
A Novel Fourier Theory on Non-linear Phases and Applications


澳门大学科技学院数学系, 澳门, 3001
Faculty of Science and Technology, University of Macau, 3001, Macau, P. R. China

收稿日期: 2018-04-17
出版日期: 2018-06-08
2018, Vol. 47(3): 321-347
DOI: 10.11845/sxjz.2018002a

[1038 KB]


摘要 信号的正频率表示自Fourier分析诞生以来一直都是物理学家、数学家以及信号分析工作者密切关注的问题. 基于调和分析和复分析方法,在过去近二十年里诞生了单分量函数理论以及基于单分量函数的函数(信号)表示理论.作为原创性理论这个方法将信号快速分解为一些具有正的非线性瞬时频率的基本信号之和.该理论植根于经典数学并可以推广到定义在高维流形上的向量值及矩阵值信号. 这从而也创立了高维空间中的有理逼近理论. 单分量函数理论包括正瞬时频率的数学定义及几个最重要的单分量函数类的刻画. 单分量函数的表示理论包括核心自适应 Fourier 分解(Core Adaptive Fourier Decomposition, 或 Core AFD) 及其若干变种,包括解绕 AFD, 循环 AFD, 再生核 Hilbert 空间的预—正交 AFD. 除了理论及方法的概述, 本文也给出了两个新证明: 迄今最一般的依据极大选择原理的自适应分解的收敛性的证明; 以及参数重复选择的及用到再生核导数的必要性的证明. 最后我们给出该理论与数学及信号分析中若干相关理论的联系, 以及该方法的某些应用.
关键词 Blaschke乘积单分量函数Hardy 空间内函数和外函数自适应Fourier分解Beurling-Lax定理再生核Hilbert空间    
Abstract:Ever since the time of Fourier, positive frequency representation of signals has always been a central problem for physicists, mathematicians and signal analysts. Based on harmonic analysis and complex analysis methods, a mono-component function theory and a function (signal) representation theory based on mono-component functions have been being established during the past two decades. Being an original theory, it rapidly decomposes a signal into a sum of certain basic signals with positive nonlinear instantaneous frequency. The theory has roots in classical mathematics and can be generalized to vector-valued and matrix-valued signals defined on high-dimensional manifolds. It also leads to establishment of rational approximation in higher dimensional spaces. The theory of mono-component functions includes the mathematical definition of positive instantaneous frequencies and description of several most important mono-component function classes. The representation theory of mono-component functions includes Core Adaptive Fourier Decomposition (or Core AFD) and several variants, including Unwinding AFD, Cyclic AFD, and Pre-orthogonal AFD in reproducing kernel Hilbert spaces. In addition to the overview of theory and methods, this paper also provides two new proofs: the most general proof so far of convergence of adaptive decomposition based on maximal selection principle; and a proof of the necessity of reproducing kernel derivatives for parameter repetition. Finally, we specify some connections between this new development with the existing pure and applied mathematics, as well as some information on applications.
Key wordsBlaschke product    mono-component    Hardy space    inner and outer functions    adaptive Fourier decomposition    Beurling-Lax theorem    reproducing kernel Hilbert space
PACS:  O1-0  
基金资助:澳门大学Multi-Year Research Grant (MYRG) (No. MYRG2016-00053-FST)和澳门政府自然和科技基金 (No. FDCT 079/2016/A2).
通讯作者: E-mail:   
[1] Alpay, D., Colombo, F., Qian, T. and Sabadini, I., Adaptive orthonormal systems for matrix-valued functions, Proc. Amer. Math. Soc., 2017, 145(5): 2089-2106.
[2] Alpay, D., Colombo, F., Qian, T. and Sabadini, I., Adaptative decomposition: The case of the Drury-Arveson space, J. Fourier Anal. Appl., 2017, 23(6): 1426-1444.
[3] Baratchart, L., Cardelli, M. and Olivi, M., Identification and rational $L^2$ approximation: a gradient algorithm, Automatica, 1991, 27(2): 413-417.
[4] Baratchart, L., Mai, W.X. and Qian, T., Greedy algorithms and rational approximation in one and several variables, In: Modern Trends in Hypercomplex Analysis (Bernstein, S., Kähler, U., Sabadini, I. and Sommen, F. eds.), Trends in Mathematics, Boston: Birkhäuser, 2016, 19-33.
[5] Bell, S.R., The Cauchy Transform, Potential Theory and Conformal Mapping, Boca Raton: CRC Press, 1992.
[6] Boashash, B., Estimating and interpreting the instantaneous frequency of a signal. 1. Fundamentals, Proc. IEEE, 1992, 80(4): 520-538.
[7] Chen, Q.H., Mai, W.X., Zhang, L.M. and Mi, W., System identification by discrete rational atoms, Automatica, 2015, 56: 53-59.
[8] Cheng, Q.S., Digital Signal Processing, Beijing: Peking University Press, 2003 (in Chinese).
[9] Cohen, L., Time-Frequency Analysis: Theory and Applications, Upper Saddle River, NJ: Prentice Hall, 1995.
[10] Coifman, R.R. and Peyriére, J., Phase unwinding, or invariant subspace decompositions of Hardy spaces, 2017, arXiv: 1707.04844.
[11] Coifman, R.R. and Steinerberger, S., Nonlinear phase unwinding of functions, J. Fourier Anal. Appl., 2017, 23(4): 778-809.
[12] Coifman, R., Steinerberger, S. and Wu, H.T., Carrier frequencies, holomorphy, and unwinding, SIAM J. Math. Anal., 2017, 49(6): 4838-4864.
[13] Colombo, F., Sabadini, I. and Sommen, F., The Fueter primitive of biaxially monogenic functions, Commun. Pure Appl. Anal., 2014, 13(2): 657-672.
[14] Colombo, F., Sabadini, I. and Sommen, F., The Fueter mapping theorem in integral form and the ${\cal F}$-functional calculus, Math. Methods Appl. Sci., 2010, 33(17): 2050-2066.
[15] Dang, P., Deng, G.T. and Qian, T., A sharper uncertainty principle, J. Funct. Anal., 2013, 265(10): 2239-2266.
[16] Dang, P., Deng, G.T. and Qian, T., A tighter uncertainty principle for linear canonical transform in terms of phase derivative, IEEE Trans. Signal Process., 2013, 61(21): 5153-5164.
[17] Dang, P., Liu, H. and Qian, T., Hilbert transformation and representation of $ax+b$ group, Canad. Math. Bull., 2018, 61: 70-84.
[18] Dang, P., Liu, H. and Qian, T., Hilbert transformation and $r\mathrm{Spin}(n)+ \mathbb{R}^n$ group, 2017, arXiv:1711.04519.
[19] Dang, P., Mai, W.X. and Qian, T., Fourier spectrum characterizations of Clifford $H^p$ spaces on $R_+^{n+1}$ for $1\leq p\leq \infty$, 2017, arXiv: 1711.02610.
[20] Dang, P. and Qian, T., Analytic phase derivatives, all-pass filters and signals of minimum phase, IEEE Trans. Signal Process., 2011, 59(10): 4708-4718.
[21] Dang, P. and Qian, T., Transient time-frequency distribution based on mono-component decompositions, Int. J. Wavelets Multiresolut. Inf. Process., 2013, 11(3): 1350022, 24 pages.
[22] Dang, P., Qian, T. and Chen, Q.H., Uncertainty principle and phase amplitude analysis of signals on the unit sphere, Adv. Appl. Clifford Al., 2017, 27(4): 2985-3013.
[23] Dang, P., Qian, T. and Yang, Y., Extra-strong uncertainty principles in relation to phase derivative for signals in Euclidean spaces, J. Math. Anal. Appl., 2016, 437(2): 912-940.
[24] Dang, P., Qian, T. and You, Z., Hardy-Sobolev spaces decomposition in signal analysis, J. Fourier Anal. Appl., 2011, 17(1): 36-64.
[25] De León, P., Beltrán, J.R. and Beltrán, F., Instantaneous frequency estimation and representation of the audio signal through Complex Wavelet Additive Synthesis, Int. J. Wavelets Multiresolut. Inf. Process., 2014, 12(3): 1450030, 28 pages.
[26] De Schepper, N., Qian, T., Sommen, F. and Wang, J.X., Holomorphic approximation of $L_2$-functions on the unit sphere in $R^3$, J. Math. Anal. Appl., 2014, 416(2): 659-671.
[27] Deng, G.T. and Qian, T., Rational approximation of functions in Hardy Spaces, Complex Anal. Oper. Th., 2016, 10(5): 903-920.
[28] Eisner, T. and Pap, M., Discrete orthogonality of the Malmquist Takenaka system of the upper half plane and rational interpolation, J. Fourier Anal. Appl., 2014, 20(1): 1-16.
[29] Falcão, M.I., Cruz, J.F. and Malonek, H.R., Remarks on the generation of monogenic functions, In: 17th International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering (Gürlebeck, K. and Könke, C. eds.), Weimar, Germany, 2006.
[30] Gabor, D., Theory of communication. Part 1: The analysis of information, J. Inst. Elect. Eng., 1946, 93(26): 429-441.
[31] Ganta, P., Manu, G. and Anil, S., New perspective for health monitoring system, Int. J. Ethics Eng. Manag. Edu., 2016, 3(10): 7-10.
[32] Garnett, J.B., Bounded Analytic Functions, New York: Academic Press, 1981.
[33] Gaudry, G.I., Long, R. and Qian, T., A martingale proof of $L^2$-boundedness of Clifford-valued singular integrals, Ann. Math. Pura Appl., 1993, 165(1): 369-394.
[34] Gaudry, G.I., Qian, T. and Wang, S.L., Boundedness of singular integral operators with holomorphic kernels on star-shaped closed Lipschitz curves, Colloq. Math., 1996, 70: 133-150.
[35] Gomes, N.R., Compressed sensing in Clifford analysis, Doctoral Dissertation, Aveiro: Universidade de Aveiro, 2015.
[36] Gorusin, G.M., 陈建功(译), 复变函数的几何理论, 上海: 上海科学技术出版社, 1956.
[37] Hummel, J.A., Multivalent starlike functions, J. d'Analyse Math., 1967, 18(1): 133-160.
[38] Kirkbas, A., Kizilkaya, A. and Bogar, E., Optimal basis pursuit based on Jaya optimization for adaptive Fourier decomposition, In: 2017 40th International Conference on Telecommunications and Signal Processing, Barcelona: IEEE, 2017: 538-543.
[39] Krausshar, R.S. and Ryan, J., Clifford and harmonic analysis on cylinders and tori, Rev. Mat. Iberoamer., 2005, 21(1): 87-110.
[40] Lei, Y., Fang, Y. and Zhang, L.M., Iterative learning control for discrete linear system with wireless transmission based on adaptive Fourier decomposition, In: 2017 36th Chinese Control Conference (CCC), Dalian: IEEE, 2017: 3343-3348.
[41] Li, C., McIntosh, A. and Qian, T., Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces, Rev. Mat. Iberoamer., 1994, 10(3): 665-721.
[42] Li, C., McIntosh, A. and Semmes, S., Convolution singular integrals on Lipschitz surfaces, J. Amer. Math. Soc., 1992, 5(3): 455-481.
[43] Li, H.C., Deng, G.T. and Qian, T., Fourier spectrum characterizations of $H^{p}$ spaces on tubes over cones for $1\leq p\leq \infty$, Complex Anal. Oper. Th., 2018, 12(5): 1193-1218.
[44] Li, H.C., Deng, G.T. and Qian, T., Hardy space decomposition of $L^{p}$ on the unit circle: $0[45] Liang, Y., Jia, L.M., Cai, G.Q. and Liu, J.Z., A new approach to diagnose rolling bearing faults based on AFD, In: Proceedings of the 2013 International Conference on Electrical and Information Technologies for Rail Transportation-Volume II, Lecture Notes in Electrical Engineering, Vol. 288, Berlin: Springer-Verlag, 2014, 573-582.
[46] Lyzzaik, A., On a conjecture of M. S. Robertson, Proc. Amer. Math. Soc., 1984, 91(1): 108-110.
[47] Mallat, S.G. and Zhang, Z.F., Matching pursuits with time-frequency dictionaries, IEEE Trans. Signal Process., 1993, 41(12): 3397-3415.
[48] Mashreghi, J. and Fricain, E., Blaschke Products and Their Applications, Boston: Springer-Verlag, 2013.
[49] McIntosh, A. and Qian, T., Convolution singular integral operators on Lipschitz curves, In: Harmonic Analysis, Lecture Notes in Math., Vol. 1494, Berlin: Springer-Verlag, 1991, 142-162.
[50] McIntosh, A. and Qian, T., Fourier multipliers along Lipschitz curves, Trans. Amer. Math. Soc., 1992, 333(1): 157-176.
[51] Mi, W. and Qian, T., Frequency-domain identification: An algorithm based on an adaptive rational orthogonal system, Automatica, 2012, 48(6): 1154-1162.
[52] Mi, W., Qian, T. and Wan, F., A fast adaptive model reduction method based on Takenaka-Malmquist systems, Syst. Control Lett., 2012, 61(1): 223-230.
[53] Mo, Y., Qian, T. and Mi, W., Sparse representation in Szegö kernels through reproducing kernel Hilbert space theory with applications, Int. J. Wavelet Multiresolut. Inf. Process., 2015, 13(4): 1550030, 22 pages.
[54] Mózes, F.E. and Szalai, J., Computing the instantaneous frequency for an ECG signal, Sci. Bull. "Petru Maior" Univ. Tîrgu Mures, 2012, 9(2): 28-32.
[55] Nahon, M., Phase Evaluation and Segmentation, Ph.D. Thesis, New Haven: Yale University, 2000.
[56] Perotti, A., Directional quaternionic Hilbert operators, In: Hypercomplex Analysis, Trends in Mathematics, Basel: Birkhäuser, 2008, 235-258.
[57] Qian, T., Singular integrals with holomorphic kernels and Fourier multipliers on star-shaped closed Lipschitz curves, Studia Math., 1997, 123(3): 195-216.
[58] Qian, T., Characterization of boundary values of functions in Hardy spaces with applications in signal analysis, J. Integral Equ. Appl., 2005, 17(2): 159-198.
[59] Qian, T., Analytic signals and harmonic measures, J. Math. Anal. Appl., 2006, 314(2): 526-536.
[60] Qian, T., Mono-components for decomposition of signals, Math. Methods Appl. Sci., 2006, 29(10): 1187-1198.
[61] Qian, T., Boundary derivatives of the phases of inner and outer functions and applications, Math. Methods Appl. Sci., 2009, 32(3): 253-263.
[62] Qian, T., Intrinsic mono-component decomposition of functions: An advance of Fourier theory, Math. Methods Appl. Sci., 2010, 33(7): 880-891.
[63] 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, 北京: 科学出版社, 2015.
[64] Qian, T., Two-dimensional adaptive Fourier decomposition, Math. Methods Appl. Sci., 2016, 39(10): 2431-2448.
[65] Qian, T., Fourier analysis on starlike Lipschitz surfaces, J. Funct. Anal., 2001, 183(2): 370-412.
[66] Qian, T., Cyclic AFD algorithm for the best rational approximation, Math. Methods Appl. Sci., 2014, 37(6): 846-859.
[67] Qian, T., Chen, Q.H. and Li, L.Q., Analytic unit quadrature signals with non-linear phase, Phys. D, 2005, 203(1/2): 80-87.
[68] Qian, T., Chen, Q.H. and Tan, L.H., Rational orthogonal systems are Schauder bases, Complex Var. Elliptic Equ., 2014, 59(6): 841-846.
[69] Qian, T., Ho, I.T., Leong, I.T. and Wang, Y.B., Adaptive decomposition of functions into pieces of non-negative instantaneous frequencies,Int. J. Wavelets Multiresolut. Inf. Process., 2010, 8(5): 813-833.
[70] Qian, T., Li, H. and Stessin, M., Comparison of adaptive mono-component decompositions, Nonlinear Anal.: Real World Appl., 2013, 14(2): 1055-1074.
[71] 钱涛, 李澎涛, Lipschitz边界上的奇异积分与Fourier理论, 北京: 科学出版社, 2017.
[72] Qian, T., Sproessig, W. and Wang, J.X., Adaptive Fourier decomposition of functions in quaternionic Hardy spaces, Math. Methods Appl. Sci., 2012, 35(1): 43-64.
[73] Qian, T. and Tan, L.H., Characterizations of mono-components: the Blaschke and starlike types, Complex Anal. Oper. Th., 2015, doi: 10.1007/s11785-015-0491-6, 17 pages.
[74] Qian, T. and Tan, L.H., Backward shift invariant subspaces with applications to band preserving and phase retrieval problems, Math. Methods Appl. Sci., 2016, 39(6): 1591-1598.
[75] Qian, T., Wang, J.X. and Yang, Y., Matching pursuits among shifted Cauchy kernels in higher-dimensional spaces, Acta Math. Sci., 2014, 34(3): 660-672.
[76] Qian, T., Wang, R., Xu, Y.S. and Zhang, H.Z., Orthonormal bases with nonlinear phases, Adv. Comput. Math., 2010, 33(1): 75-95.
[77] Qian, T. and Wang, Y.B., Adaptive Fourier series---A variation of greedy algorithm, Adv. Comput. Math., 2011, 34(3): 279-293.
[78] Qian, T. and Wegert, E., Optimal approximation by Blaschke forms, Complex Var. Elliptic Equ., 2013, 58(1): 123-133.
[79] Qian, T., Xu, Y.S., Yan, D.Y., Yan, L.X. and Yu, B., Fourier spectrum characterization of Hardy spaces and applications, Proc. Amer. Math. Soc., 2009, 137(3): 971-980.
[80] Qian, T. and Yang, Y., Hilbert transforms on the sphere with the Clifford algebra setting, J. Fourier Anal. Appl., 2009, 15(6): 753-774.
[81] Qian, T., Zhang, L.M. and Li, Z.X. Algorithm of adaptive Fourier decomposition, IEEE Trans. Signal Process., 2011, 59(12): 5899-5906.
[82] Qu, W. and Dang, P., Rational approximation in the Bergman spaces, 2018, arXiv: 1803.04609.
[83] Sakaguchi, F. and Hayashi, M., Integer-type algorithm for higher order differential equations by smooth wavepackets, small Part I: Mathematical framework, 2009, arXiv: 0903.4848.
[84] Sakaguchi, F. and Hayashi, M., Differentiability of eigenfunctions of the closures of differential operators with polynomial-type coefficients, 2009, arXiv: 0903.4852.
[85] Sakaguchi, F. and Hayashi, M., General theory for integer-type algorithm for higher order differential equations, Numer. Funct. Anal. Opt., 2011, 32(5): 541-582.
[86] Sakaguchi, F. and Hayashi, M., Practical implementation and error bound of integer-type algorithm for higher-order differential equations, Numer. Funct. Anal. Opt., 2011, 32(12): 1316-1364.
[87] Salomon, L., Analyse de l'anisotropie dans des images texturées, Rapport---Introduction à la Recherche en Laboratoire, 2016 (in French).
[88] Stein, E.M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton: Princeton University Press, 1971.
[89] Tan, L.H., Shen, L.X. and Yang, L.H., Rational orthogonal bases satisfying the Bedrosian Identity, Adv. Comput. Math., 2010, 33(3): 285-303.
[90] Tan, L.H. and Qian, T., Extracting outer function part from Hardy space function, Sci. China Math., 2017, 60(11): 2321-2336.
[91] Tan, L.H., Qian, T. and Chen, Q.H., New aspects of Beurling-Lax shift invariant subspaces, Appl. Math. Comput., 2015, 256: 257-266.
[92] Tan, L.H., Yang, L.H. and Huang, D.R., The structure of instantaneous frequencies of periodic analytic signals, Sci. China Math., 2010, 53(2): 347-355.
[93] Temlyakov, V.N., Greedy algorithm and $m$-term trigonometric approximation, Constr. Approx., 1998, 14(4): 569-587.
[94] Vatchev, V., A class of intrinsic trigonometric mode polynomials, In: International Conference Approximation Theory, Cham: Springer, 2017, 361-373.
[95] Vliet, D.V., Analytic signals with nonnegative instantaneous frequency, J. Integral Equ. Appl., 2009, 21(1): 95-111.
[96] Walsh, J.L., Interpolation and Approximation by Rational Functions in the Complex Plane, Providence, RI: American Mathematical Society, 1969.
[97] Wang, J.X. and Qian, T., Approximation of monogenic functions by higher order Szegö kernels on the unit ball and half space, Sci. China Math., 2014, 57(9): 1785-1797.
[98] Wang, S.L., Simple proofs of the Bedrosian equality for the Hilbert transform, Sci. China Ser. A: Math., 2009, 52(3): 507-510.
[99] Wang, Z., Da Cruz, J.N. and Wan, F., Adaptive Fourier decomposition approach for lung-heart sound separation, In: 2015 IEEE International Conference on Computational Intelligence and Virtual Environments for Measurement Systems and Applications (CIVEMSA), Shenzhen: IEEE, 2015.
[100] Weiss, M. and Weiss, G., A derivation of the main results of the theory of $H^p$ spaces, Rev. Un. Mat. Argentina, 1962, 20: 63-71.
[101] 武伟, 八元数分析和钱方法在数字图像处理中的应用研究, 硕士论文, 广州: 华南师范大学, 2014.
[102] Wu, M.Z., Wang, Y. and Li, X.M., Fast implementation of two dimensional Qian method and its application in digital watermarking, Computer Engineering and Design, 2016, 37(11): 3136-3140 (in Chinese).
[103] Wu, M.Z., Wang, Y. and Li, X.M., An improved two dimensional Qian method and its application of image denoising, J. South China Normal Univ. Nat. Sci. Ed., 2016, 48(4): 119-124 (in Chinese).
[104] Yang, Y., Qian, T. and Sommen, F., Phase derivative of monogenic signals in higher dimensional spaces, Complex Anal. Oper. Th., 2012, 6(5): 987-1010.
[105] Yu, B. and Zhang, H.Z., The Bedrosian identity and homogeneous semi-convolution equations, J. Integral Equ. Appl., 2008, 20(4): 527-568.
[106] Zhang, L.M., A new time-frequency speech analysis approach based on adaptive Fourier decomposition, Internat. Science Index, Electrical Computer Engineering, 2013, 7(7): 938-942.
[107] Zhang, L.M., Liu, N. and Yu, P.Y., A novel instantaneous frequency algorithm and its application in stock index movement prediction, IEEE J. Sel. Top. Signal Process., 2012, 6(4): 311-318.
[108] Zhang, L.M., Qian, T., Mai, W.X. and Dang, P., Adaptive Fourier decomposition-based Dirac type time-frequency distribution, Math. Methods Appl. Sci., 2017, 40(8): 2815-2833.
No related articles found!
Full text



首页 · 关于 · 关于OA · 法律公告 · 收录须知 · 联系我们 · 注册 · 登录

© 2015-2017 北京大学图书馆 .