The AFD methods to compute Hilbert transform | |
Mo Y.1; Qian T.1; Mai W.1; Chen Q.2 | |
2015 | |
Source Publication | Applied Mathematics Letters |
ISSN | 18735452 08939659 |
Volume | 45Pages:18-24 |
Abstract | 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 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. |
Keyword | Adaptive Fourier Decomposition Hardy Space Hilbert Transform Orthogonal Rational System Takenaka-malmquist System |
DOI | http://doi.org/10.1016/j.aml.2014.12.017 |
URL | View the original |
Indexed By | SCI |
Language | 英语 |
WOS Research Area | Mathematics |
WOS Subject | Mathematics, Applied |
WOS ID | WOS:000351976300004 |
Fulltext Access | |
Citation statistics | |
Document Type | Journal article |
Collection | DEPARTMENT OF MATHEMATICS |
Corresponding Author | Mo Y. |
Affiliation | 1.Department of Mathematics, University of Macau, Macao 2.Cisco School of Informatics, Guangdong University of Foreign Studies, China |
First Author Affilication | University of Macau |
Corresponding Author Affilication | University of Macau |
Recommended Citation GB/T 7714 | Mo Y.,Qian T.,Mai W.,et al. The AFD methods to compute Hilbert transform[J]. Applied Mathematics Letters,2015,45:18-24. |
APA | Mo Y.,Qian T.,Mai W.,&Chen Q..(2015).The AFD methods to compute Hilbert transform.Applied Mathematics Letters,45,18-24. |
MLA | Mo Y.,et al."The AFD methods to compute Hilbert transform".Applied Mathematics Letters 45(2015):18-24. |
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