|
|
Positive-instantaneous frequency and approximation |
Tao QIAN() |
Macao Center for Mathematical Science, Macao University of Science and Technology, Macao, China |
|
|
Abstract 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 one can associate it with a Hardy space function whose real part coincides with . Such function has the form , where stands for the Hilbert transformation of the context. We develop fast converging expansions of in orthogonal terms of the form where ’s are also Hardy space functions but with the additional properties The original real-valued function is accordingly expanded which, besides the properties of and given above, also satisfies Real-valued functions that satisfy the condition are called mono-components. If is a mono-component, then the phase derivative is defined to be instantaneous frequency of . 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.
|
Keywords
Möbius transform
blaschke product
mono-component
hilbert transform
hardy space
inner and outer functions
adaptive fourier decomposition
rational orthogonal system
nevanlinna factorization
beurling-lax theorem
reproducing kernel hilbert space
several complex variables
clifford algebra
pre-orthogonal AFD
|
Corresponding Author(s):
Tao QIAN
|
Issue Date: 25 May 2022
|
|
1 |
D Alpay, F Colombo, T Qian, I. Sabadini Adaptive orthonormal systems for matrix-valued functions. Proceedings of the American Mathematical Society, 2017, 145(5): 2089–2106
https://doi.org/10.1090/proc/13359
|
2 |
D Alpay, F Colombo, T Qian, I. Sabadini Adaptive Decomposition: The Case of the Drury-Arveson Space. Journal of Fourier Analysis and Applications, 2017, 23(6), 1426–1444
https://doi.org/10.1007/s00041-016-9508-4
|
3 |
A Axelsson, K I Kou, T. Qian Hilbert transforms and the Cauchy integral in Euclidean space. Studia Mathematica, 2009, 193(2): 161–187
https://doi.org/10.4064/sm193-2-4
|
4 |
L Baratchart, M Cardelli, M. Olivi Identification and rational L2 approximation, a gradient algorithm. Automatica, 1991, 27: 413–418
https://doi.org/10.1016/0005-1098(91)90092-G
|
5 |
L Baratchart, P Dang, T. Qian Hardy-Hodge Decomposition of Vector Fields in Rn. Transactions of the American Mathematical Society, 2018, 370: 2005–2022
https://doi.org/10.1090/tran/7202
|
6 |
L Baratchart, W X Mai, T. Qian Greedy Algorithms and Rational Approximation in One and Several Variables. In: Bernstein S., Kaehler U., Sabadini I., Sommen F. (eds) Modern Trends in Hypercomplex Analysis. Trends in Mathematics, 2016: 19–33
https://doi.org/10.1007/978-3-319-42529-0_2
|
7 |
S. Bell The Cauchy Transform, Potential theory and Conformal Mappings. CRC Press, Boca, Raton, 1992
|
8 |
B. Boashash Estimating and interpreting the instantaneous frequency of a signal-Part 1: Fundamentals. Proceedings of The IEEE, 1992, 80(4): 520–538
https://doi.org/10.1109/5.135376
|
9 |
J R Beltrán, P. de León Instantaneous frequency estimation and representation of the audio signal through Complex Wavelet Additive Synthesis. International Journal of Wavelets, Multiresolution and Information Processing, 2014, 12(03): 1450030
https://doi.org/10.1142/S0219691314500301
|
10 |
M T Cheng, G T. Deng Lecture notes on harmonic analysis. Peking University, 1979
|
11 |
R Coifman, J. Peyriére Phase unwinding, or invariant subspace decompositions of Hardy spaces. Journal of Fourier Analysis and Applications, 2019, 25: 684–695
https://doi.org/10.1007/s00041-018-9623-5
|
12 |
R Coifman, S. Steinerberger Nonlinear phase unwinding of functions. J Fourier Anal Appl, 2017, 23: 778–809
https://doi.org/10.1007/s00041-016-9489-3
|
13 |
R Coifman, S Steinerberger, H T. Wu Carrier frequencies, holomorphy and unwinding. SIAM J. Math. Anal., 2017, 49(6): 4838–4864
https://doi.org/10.1137/16M1081087
|
14 |
Q S. Cheng Digital Signal Processing. Peking University Press, 2003, in Chinese
|
15 |
Q H Chen, T Qian, L H. Tan Constructive Proof of Beurling-Lax Theorem, Chin. Ann. of Math., 2015, 36: 141–146
https://doi.org/10.1007/s11401-014-0870-8
|
16 |
L. Cohen Time-Frequency Analysis: Theory and Applications. Prentice Hall, 1995
|
17 |
Q H Chen, W X Mai, L M Zhang, W. Mi System identification by discrete rational atoms. Automatica, 2015, 56: 53–59
https://doi.org/10.1016/j.automatica.2015.03.022
|
18 |
F Colombo, I Sabadini, F. Sommen The Fueter primitive of bi-axially monogenic functions. Communications on Pure and Applied Analysis, 2014, 13(2): 657–672
https://doi.org/10.3934/cpaa.2014.13.657
|
19 |
F Colombo, I Sabadini, F. Sommen The Fueter mapping theorem in integral form and the ℱ-functional calculus. Mathematical Methods in the Applied Sciences, 2010, 33(17): 2050–2066
https://doi.org/10.1002/mma.1315
|
20 |
P Dang, G T Deng, T. Qian A Sharper Uncertainty principle. Journal of Functional Analysis, 2013, 265(10): 2239–2266
https://doi.org/10.1016/j.jfa.2013.07.023
|
21 |
P Dang, G T Deng, T. Qian A Tighter Uncertainty Principle For Linear Canonical Transform in Terms of Phase Derivative. IEEE Transactions on Signal Processing, 2013, 61(21): 5153–5164
https://doi.org/10.1109/TSP.2013.2273440
|
22 |
P Dang, H Liu, T. Qian Hilbert Transformation and Representation of ax + b Group. Canadian Mathematical Bulletin, 2018, 61(1): 70–84
https://doi.org/10.4153/CMB-2017-063-0
|
23 |
P Dang, H Liu, T. Qian Hilbert Transformation and rSpin(n)+Rn Group. arXiv:1711. 04519v1[math.CV], 2017
|
24 |
G Davis, S Mallet, M. Avellaneda Adaptive Greedy Approximations, Constr. Approxi., 1997, 13: 57–98
https://doi.org/10.1007/BF02678430
|
25 |
P Dang, W X Mai, T. Qian Fourier Spectrum Characterizations of Clifford Hp Spaces on R+n+1 for 1 ≤ p ≤ ∞. Journal of Mathematical Analysis and Applications, 2020, 483: 123598
https://doi.org/10.1016/j.jmaa.2019.123598
|
26 |
P Dang, T. Qian Analytic Phase Derivatives, All-Pass Filters and Signals of Minimum Phase. IEEE Transactions on Signal Processing, 2011, 59(10): 4708–4718
https://doi.org/10.1109/TSP.2011.2160260
|
27 |
P Dang, T. Qian Transient Time-Frequency Distribution based on Mono-component Decompositions. International Journal of Wavelets, Multiresolution and Information Processing, 2013, 11(3): 1350022
https://doi.org/10.1142/S0219691313500227
|
28 |
P Dang, T Qian, Q H. Chen Uncertainty Principle and Phase Amplitude Analysis of Signals on the Unit Sphere. Advances in Applied Clifford Algebras, 2017, 27(4): 2985–3013
https://doi.org/10.1007/s00006-017-0808-9
|
29 |
P Dang, T Qian, Z. You Hardy-Sobolev spaces decomposition and applications in signal analysis. J. Fourier Anal. Appl., 2011. 17(1): 36–64
https://doi.org/10.1007/s00041-010-9132-7
|
30 |
P Dang, T Qian, Y. Yang Extra-strong uncertainty principles in relation to phase derivative for signals in Euclidean spaces. Journal of Mathematical Analysis and Applications, 2016, 437(2): 912–940
https://doi.org/10.1016/j.jmaa.2016.01.039
|
31 |
G T Deng, T. Qian Rational approximation of Functions in Hardy Spaces. Complex Analysis and Operator Theory, 2016, 10(5): 903–920
https://doi.org/10.1007/s11785-015-0490-7
|
32 |
T Eisner, M. Pap Discrete orthogonality of the Malmquist Takenaka system of the upper half plane and rational interpolation. Journal of Fourier Analysis and Applications, 2014, 20(1): 1–16
https://doi.org/10.1007/s00041-013-9285-2
|
33 |
M I Falcão, J F Cruz, H R. Malonek Remarks on the generation of monogenic functions. International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering, 17, Weimar, 2006
|
34 |
P Fulcheri, P. Olivi Matrix rational H2 approximation: a gradient algorithm based on schur analysis. SIAM I. Control Optim., 1998, 36(6): 2103–2127
https://doi.org/10.1137/S0363012995284230
|
35 |
D. Gabor Theory of communication. J. IEE., 1946, 93: 429–457
https://doi.org/10.1049/ji-3-2.1946.0074
|
36 |
G I Gaudry, R Long, T. Qian A Martingale proof of L2-boundednessof Clifford-Valued Singular Integrals. Annali di Mathematica Pura Ed Applicata, 1993, 165: 369–394
https://doi.org/10.1007/BF01765857
|
37 |
G Gaudry, T Qian, S L Wang, Boundedness of singular integrals with holomorphic kernels on star-shaped closed Lipschitz curves. Colloquium Mathematicum, 1996: 133–150
https://doi.org/10.4064/cm-70-1-133-150
|
38 |
J B. Garnett Bounded Analyic Functions. Academic Press, 1981
|
39 |
N R. Gomes Compressive sensing in Clifford analysis. Doctoral Dissertation, Universidade de Aveiro (Portugal), 2015
|
40 |
S. Gong Private comminication, 2002
|
41 |
G M. Gorusin Geometrical Theory of Functions of One Complex Variable. translated by Jian-Gong Chen, 1956
|
42 |
P Ganta, G Manu, S. Anil Sooram New Perspective for Health Monitoring System. International Journal of Ethics in Engineering and Management Education, 2016
|
43 |
J A. Hummel Multivalent starlike function. J. d’ analyse Math., 1967, 18: 133–160
https://doi.org/10.1007/BF02798041
|
44 |
N E. Huang The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lndon, 1998: 903–995
https://doi.org/10.1098/rspa.1998.0193
|
45 |
A Kirkbas, A Kizilkaya, E. Bogar Optimal basis pursuit based on jaya optimization for adaptive fourier decomposition. Telecommunications and Signal Processing, 2017 40th International Conference on IEEE: 538–543
https://doi.org/10.1109/TSP.2017.8076045
|
46 |
R S Krausshar, J. Ryan Clifford and harmonic analysis on cylinders and tori. Revista Matematica Iberoamericana, 2005, 21(1): 87–110
https://doi.org/10.4171/RMI/416
|
47 |
H C Li, G T Deng, T. Qian Fourier Spectrum Characterizations of Hp Spaces on Tubes Over Cones for 1 ≦ p ≦ ∞. Complex Analysis and Operator Theory, 2018, 12: 1193–1218
https://doi.org/10.1007/s11785-017-0737-6
|
48 |
H C Li, G T Deng, T. Qian Hardy space decomposition of on the unit circle: 0¡p¡1. Complex Variables and Elliptic Equations: An International Journal, 2016, 61(4): 510–523
https://doi.org/10.1080/17476933.2015.1102901
|
49 |
Y Lei, Y Fang, L M. Zhang Iterative learning control for discrete linear system with wireless transmission based on adaptive fourier decomposition. Control Conference (CCC), 2017 36th Chinese IEEE
https://doi.org/10.23919/ChiCC.2017.8027875
|
50 |
Y Liang, L M Jia, G. Cai A new approach to diagnose rolling bearing faults based on AFD. Proceedings of the 2013 International Conference on Electrical and Information Technologies for Rail Transportation-Volume II, Springer
https://doi.org/10.1007/978-3-642-53751-6_61
|
51 |
C Li, A McIntosh, T. Qian Clifford algebras, Fourier transforms, and singular Convolution operators on Lipschitz surfaces. Revista Matematica Iberoamericana, 1994, 10(3): 665–695
https://doi.org/10.4171/RMI/164
|
52 |
C Li, A McIntosh, S. Semmes Convolution Singular Integrals on Lipschitz Surfaces. Journal of the American Mathematical Society, 1992: 455–481
https://doi.org/10.1090/S0894-0347-1992-1157291-5
|
53 |
S Li, T Qian, W X. Mai Sparse Reconstruction of Hardy Signal And Applications to Time-Frequency Distribution. International Journal of Wavelets, Multiresolution and Information Processing, 2013
|
54 |
A. Lyzzaik On a conjecture of M.S. Robertson. Proc. Am. Math. Soc., 1984, 91: 108–210
https://doi.org/10.1090/S0002-9939-1984-0735575-7
|
55 |
A McIntosh, T. Qian Convolution singular integrals on Lipschitz curves. Springer-Verlag, Lecture Notes in Maths, 1991, 1494: 142–162
https://doi.org/10.1007/BFb0087766
|
56 |
A McIntosh, T. Qian Lp Fourier multipliers along Lipschitz curves. Transactions of The American Mathematical Society, 1992, 333(1): 157–176
https://doi.org/10.1090/S0002-9947-1992-1062194-7
|
57 |
J Mashreghi, E. Fricain Blaschke products and their applications. Springer, 2013
https://doi.org/10.1007/978-1-4614-5341-3
|
58 |
W X Mai, T Qian, S. Saitoh Adaptive Decomposition of Functions with Reproducing Kernels. in preparation
|
59 |
W Mi, T. Qian Frequency Domain Identification: An Algorithm Based On Adaptive Rational Orthogonal System. Automatica, 2012, 48(6): 1154–1162
https://doi.org/10.1016/j.automatica.2012.03.002
|
60 |
Y Mo, T Qian, W. Mi Sparse Representation in Szego Kernels through Reproducing Kernel Hilbert Space Theory with Applications. International Journal of Wavelet. Multiresolution and Information Processing, 2015, 13(4): 1550030
https://doi.org/10.1142/S0219691315500307
|
61 |
W Mi, T Qian, F. Wan A Fast Adaptive Model Reduction Method Based on Takenaka-Malmquist Systems. Systems and Control Letters, 2012, 61(1): 223–230
https://doi.org/10.1016/j.sysconle.2011.10.016
|
62 |
F E Mozes, J. Szalai Computing the instantaneous frequency for an ECG signal. Scientific Bulletin of the “Petru Maior” University of Targu Mures, 2012, 9(2): 28
|
63 |
M. Nahon Phase Evaluation and Segmentation. Ph.D. Thesis, Yale University, 2000
|
64 |
A. Perotti Directional quaternionic Hilbert operators. Hypercomplex analysis, Birkhüser Basel, 2008: 235–258
https://doi.org/10.1007/978-3-7643-9893-4_15
|
65 |
B. Picinbono On instantaneous amplitude and phase of signals. IEEE Transactions on Signal Processing, 1997, 45(3): 552–560
https://doi.org/10.1109/78.558469
|
66 |
T. Qian Singular integrals with holomorphic kernels and Fourier multipliers on starshape Lipschitz curves. Studia Mathematica, 1997, 123(3): 195–216
https://doi.org/10.4064/sm-123-3-195-216
|
67 |
T. Qian Characterization of boundary values of functions in Hardy spaces with applications in signal analysis. Journal of Integral Equations and Applications, 2005, 17(2): 159–198
https://doi.org/10.1216/jiea/1181075323
|
68 |
T. Qian Analytic Signals and Harmonic Measures. Journal of Mathematical Analysis and Applications, 2006, 314(2): 526–536
https://doi.org/10.1016/j.jmaa.2005.04.003
|
69 |
T. Qian Mono-components for decomposition of signals. Mathematical Methods in the Applied Sciences, 2006, 29(10): 1187–1198
https://doi.org/10.1002/mma.721
|
70 |
T. Qian Boundary Derivatives of the Phases of Inner and Outer Functions and Applications. Mathematical Methods in the Applied Sciences, 2009, 32: 253–263
https://doi.org/10.1002/mma.1032
|
71 |
T. Qian Intrinsic mono-component decomposition of functions: An advance of Fourier theory. Mathematical Methods in Applied Sciences, 2010, 33: 880–891
https://doi.org/10.1002/mma.1214
|
72 |
T. Qian Two-Dimensional Adaptive Fourier Decomposition. Mathematical Methods in the Applied Sciences, 2016, 39(10): 2431–2448
https://doi.org/10.1002/mma.3649
|
73 |
T. Qian Adaptive Fourier Decomposition: A Mathematical Method Through Complex Analysis. Harmonic Analysis and Signal Analysis, Science Press (in Chinese), 2015
|
74 |
T Qian, P T. Li Singular Integrals and Fourier Theory. Science Press (in Chinese), 2017
|
75 |
T. Qian Fourier analysis on starlike Lipschitz surfaces. Journal of Functional Analysis, 2001, 183: 370–412
https://doi.org/10.1006/jfan.2001.3750
|
76 |
T. Qian Cyclic AFD Algorithm for Best Approximation by Rational Functions of Given Order. Mathematical Methods in the Applied Sciences, 2014, 37(6): 846–859
https://doi.org/10.1002/mma.2843
|
77 |
T Qian, Q H Chen, L H. Tan Rational Orthogonal Systems are Schauder Bases. Complex Variables and Elliptic Equations, 2014, 59(6): 841–846
https://doi.org/10.1080/17476933.2013.787532
|
78 |
T Qian, Q H Chen, L Q. Li Analytic unit quadrature signals with non-linear phase. Physica D: Nonlinear Phenomena, 2005, 303: 80–87
https://doi.org/10.1016/j.physd.2005.03.005
|
79 |
T Qian, J S. Huang AFD on the n-Torus. in preparation
|
80 |
T Qian, I T Ho, I T Leong, Y B. Wang Adaptive decomposition of functions into pieces of non-negative instantaneous frequencies. International Journal of Wavelets, Multiresolution and Information Processing, 2010, 8(5): 813–833
https://doi.org/10.1142/S0219691310003791
|
81 |
T Qian, H Li, M. Stessin Comparison of Adaptive Mono-component Decompositions. Nonlinear Analysis: Real World Applications, 2013, 14(2): 1055–1074
https://doi.org/10.1016/j.nonrwa.2012.08.017
|
82 |
T Qian, L H. Tan Characterizations of Mono-components: the Blaschke and Starlike types. Complex Analysis and Operator Theory, 2015: 1–17
https://doi.org/10.1007/s11785-015-0491-6
|
83 |
T Qian, L H. Tan Backward shift invariant subspaces with applications to band preserving and phase retrieval problems. Mathematical Methods in the Applied Sciences, 2016, 39(6): 1591–1598
https://doi.org/10.1002/mma.3591
|
84 |
T Qian, Y B. Wang Adaptive Fourier Series-A Variation of Greedy Algorithm. Advances in Computational Mathematics, 2011, 34(3): 279–293
https://doi.org/10.1007/s10444-010-9153-4
|
85 |
T Qian, E. Wegert Optimal Approximation by Blaschke Forms. Complex Variables and Elliptic Equations, 2013, 58(1): 123–133
https://doi.org/10.1080/17476933.2011.557152
|
86 |
T Qian, J X. SproessigW,Wang Adaptive Fourier decomposition of functions in quaternionic Hardy spaces. Mathematical Methods in the Applied Sciences, 2012, 35(1): 43–64
https://doi.org/10.1002/mma.1532
|
87 |
T Qian, L H Tan, Y B. Wang Adaptive Decomposition by Weighted Inner Functions: A Generalization of Fourier Serie. J. Fourier Anal. Appl., 2011, 17(2): 175–190
https://doi.org/10.1007/s00041-010-9154-1
|
88 |
T Qian, J X. Wang Adaptive Decomposition of Functions by Higher Order Szegö Kernels I: A Method for Mono-component Decomposition. submitted to Acta Applicanda Mathematicae
|
89 |
T Qian, J Z. Wang Gradient Descent Method for Best Blaschke-Form Approximation of Function in Hardy Space. http://arxiv.org/abs/1803.08422
|
90 |
T Qian, J X. SproessigW,Wang Adaptive Fourier decomposition of functions in quaternionic Hardy spaces. Mathematical Methods in the Applied Sciences, 2012, 35: 43–64
https://doi.org/10.1002/mma.1532
|
91 |
T Qian, J X Wang, Y. Yang Matching Pursuits among Shifted Cauchy Kernels in Higher-Dimensional Spaces. Acta Mathematica Scientia, 2014, 34(3): 660–672
https://doi.org/10.1016/S0252-9602(14)60038-2
|
92 |
T Qian, R Wang, Y S Xu, H Z. Zhang Orthonormal Bases with Nonlinear Phase. Advances in Computational Mathematics, 2010, 33: 75–95
https://doi.org/10.1007/s10444-009-9120-0
|
93 |
T Qian, Y S Xu, D Y Yan, L X Yan, B. Yu Fourier Spectrum Characterization of Hardy Spaces and Applications. Proceedings of the American Mathematical Society, 2009, 137(3): 971–980
https://doi.org/10.1090/S0002-9939-08-09544-0
|
94 |
T Qian, Y. Yang Hilbert Transforms on the Sphere With the Clifford Algebra Setting. Journal of Fourier Analysis and Applications, 2019, 15: 753–774
https://doi.org/10.1007/s00041-009-9062-4
|
95 |
T. Qian L M Zhang, Z X. Li Algorithm of Adaptive Fourier Decomposition. IEEE Transaction on Signal Processing, Dec., 2011, 59(12): 5899–5902
https://doi.org/10.1109/TSP.2011.2168520
|
96 |
W Qu, P. Dang Rational Approximation in a Class of Weighted Hardy Spaces. Complex Analysis and Operator Theory volume, 2019, 13: 1827–1852
https://doi.org/10.1007/s11785-018-0862-x
|
97 |
L. Salomon Analyse de l’anisotropie dans des images texturées, 2016
|
98 |
F Sakaguchi, M. Hayashi General theory for integer-type algorithm for higher order differential equations. Numerical Functional Analysis and Optimization, 2011, 32(5): 541–582
https://doi.org/10.1080/01630563.2011.557917
|
99 |
F Sakaguchi, M. Hayashi Differentiability of eigenfunctions of the closures of differential operators with rational coefficient functions, arXiv:0903.4852, 2009
|
100 |
F Sakaguchi, M. Hayashi Practical implementation and error bound of integer-type algorithm for higher-order differential equations. Numerical Functional Analysis and Optimization, 2011, 32(12): 1316–1364
https://doi.org/10.1080/01630563.2011.595602
|
101 |
F Sakaguchi, M. Hayashi Integer-type algorithm for higher order differential equations by smooth wavepackets. arXiv:0903.4848, 2009
|
102 |
D Schepper, T Qian, F Sommen, J X. Wang Holomorphic Approximation of L2-functions on the Unit Sphere in R3. Journal of Mathematical Analysis and Applications, 2014, 416(2): 659–671
https://doi.org/10.1016/j.jmaa.2014.02.065
|
103 |
R C Sharpley, V. Vatchev Analysis of intrinsic mode functions. Constructive Approximation, 2006, 24: 17–47
https://doi.org/10.1007/s00365-005-0603-z
|
104 |
E M Stein, G. Weiss Introduction to Fourirer Analysis on Euclidean Spaces. Princeton University Press, Princeton, New Jersey, 1971
|
105 |
L H Tan, L X Shen, L H. Yang Rational orthogonal bases satisfying the Bedrosian Identity. Advances in Computational Mathematics, 2010, 33: 285–303
https://doi.org/10.1007/s10444-009-9133-8
|
106 |
L H Tan, L H Yang, D R. Huang The structure of instantaneous frequencies of periodic analytic signals. Sci. China Math., 2010, 53(2): 347–355
https://doi.org/10.1007/s11425-009-0093-8
|
107 |
L H Tan, T. Qian Backward Shift Invariant Subspaces With Applications to Band Preserving and Phase Retrieval Problems
|
108 |
L H Tan, T. Qian Extracting Outer Function Part from Hardy Space Function. Science China Mathematics, 2017, 60(11): 2321–2336
https://doi.org/10.1007/s11425-017-9169-5
|
109 |
L H Tan, T Qian, Q H. Chen New aspects of Beurling Lax shift invariant subspaces. Applied Mathematics and Computation, 2015: 257–266
https://doi.org/10.1016/j.amc.2014.12.147
|
110 |
V. Vatchev A class of intrinsic trigonometric mode polynomials. International Conference Approximation Theory. Springer, Cham, 2016: 361–373
https://doi.org/10.1007/978-3-319-59912-0_18
|
111 |
D V. Vliet Analytic signals with non-negative instantaneous frequency. Journal of Integral Equations and Applications, 2009, 21: 95–111
https://doi.org/10.1216/JIE-2009-21-1-95
|
112 |
J L. Walsh Interpolation and Approximation by Rational Functions in the Complex Plane. American Mathematical Society: Providence, RI, 1969
|
113 |
S L. Wang Simple Proofs of the Bedrosian Equality for the Hilbert Transform. Science in China, Series A: Mathematics, 2009, 52(3): 507–510
https://doi.org/10.1007/s11425-009-0001-2
|
114 |
J X Wang, T. Qian Approximation of monogenic functions by higher order Szegö kernels on the unit ball and the upper half space. Sciences in China: Mathematics, 2014, 57(9): 1785–1797
https://doi.org/10.1007/s11425-013-4710-1
|
115 |
G Weiss, M. Weiss A derivation of the main results of the theory of Hp-spaces. Rev. Un. Mat. Argentina, 1962, 20: 63–71
|
116 |
W. Wu Applications in Digital Image Processing of Octonions Analysis and the Qian Method. South China Normal University, 2014
|
117 |
Z Wang, J N da Cruz, F. Wan Adaptive Fourier decomposition approach for lung-heart sound separation. Computational Intelligence and Virtual Environments for Measurement Systems and Applications (CIVEMSA). 2015 IEEE International Conference on. IEEE, 2015
https://doi.org/10.1109/CIVEMSA.2015.7158631
|
118 |
M Z Wu, Y Wang, X M. Li Fast Algorithm of The Qian Method in Digital Watermarking. Computer Engineering and Desining, 2016
|
119 |
M Z Wu, Y Wang, X M. Li Improvement of 2D Qian Method and its Application in Image Denoising. South China Normal University, 2016
|
120 |
Y S. Xu Private comminication, 2005
|
121 |
Y Yang, T Qian, F. Sommen Phase Derivative of Monogenic Signals in Higher Dimensional Spaces. Complex Analysis and Operator Theory, 2012, 6(5): 987–1010
https://doi.org/10.1007/s11785-011-0210-x
|
122 |
B Yu, H Z. Zhang The Bedrosian Identity and Homogeneous Semi-convolution Equations. Journal of Integral Equations and Applications, 2008, 20: 527–568
https://doi.org/10.1216/JIE-2008-20-4-527
|
123 |
L. Zhang A New Time-Frequency Speech Analysis Approach Based On Adaptive Fourier Decomposition. World Academy of Science, Engineering and Technology, International Journal of Electrical, Computer, Energetic, Electronic and Communication Engineering, 2013
|
124 |
L M Zhang, N Liu, P. Yu A novel instantaneous frequency algorithm and its application in stock index movement prediction. IEEE Journal of Selected Topics in Signal Processing, 2012, 6(4): 311–318
https://doi.org/10.1109/JSTSP.2012.2199079
|
125 |
L M Zhang, T Qian, W X Mai, P. Dang Adaptive Fourier decomposition-based Dirac type time-frequency distribution. Mathematical Methods in the Applied Sciences, 2017, 40(8): 2815–2833
https://doi.org/10.1002/mma.4199
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
|
Shared |
|
|
|
|
|
Discussed |
|
|
|
|