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The AFD methods to compute Hilbert transform
Mo Y.1; Qian T.1; Mai W.1; Chen Q.2
Source PublicationApplied Mathematics Letters
ISSN18735452 08939659

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.

KeywordAdaptive Fourier Decomposition Hardy Space Hilbert Transform Orthogonal Rational System Takenaka-malmquist System
URLView the original
Indexed BySCI
WOS Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000351976300004
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Cited Times [WOS]:5   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
Corresponding AuthorMo Y.
Affiliation1.Department of Mathematics, University of Macau, Macao
2.Cisco School of Informatics, Guangdong University of Foreign Studies, China
First Author AffilicationUniversity of Macau
Corresponding Author AffilicationUniversity 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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