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Cyclic AFD algorithm for best rational approximation
 Math. Meth. Appl. Sci
"... We propose a practical algorithm of best rational approximation of a given order to a function in the Hardy H2 space on the unit circle or on the real line. The type approximation is proved to be equivalent with Blaschke form approximation. The algorithm is called Cyclic AFD as it adaptively selects ..."
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We propose a practical algorithm of best rational approximation of a given order to a function in the Hardy H2 space on the unit circle or on the real line. The type approximation is proved to be equivalent with Blaschke form approximation. The algorithm is called Cyclic AFD as it adaptively selects one parameter during each cycle based on the Maximal Selection Principle used in adaptive Fourier decomposition (AFD).
International Journal of Wavelets, Multiresolution and Information Processing c ⃝ World Scientic Publishing Company ADAPTIVE FOURIER DECOMPOSITION AND RATIONAL APPROXIMATION{PART II: SOFTWARE SYSTEM DESIGN AND DEVELOPMENT
, 2014
"... Communicated by (xxxxxxxxxx) This paper proposes a new software system which provides algorithms for three variations of adaptive Fourier decomposition (AFD), including Core AFD, Cyclic AFD and Unwending AFD. The related time frequency distributions are also provided. Remarks are made for the algor ..."
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Communicated by (xxxxxxxxxx) This paper proposes a new software system which provides algorithms for three variations of adaptive Fourier decomposition (AFD), including Core AFD, Cyclic AFD and Unwending AFD. The related time frequency distributions are also provided. Remarks are made for the algorithms design and development. The software system can be used to analyze any signal with nite energy.
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"... In the literature adaptive Fourier decomposition is abbreviated as AFD that addresses adaptive rational approximation, or alternatively adaptive TakenakaMalmquist system approximation. The AFD type approximations may be characterized as adaptive approximations by linear combinations of parameterize ..."
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In the literature adaptive Fourier decomposition is abbreviated as AFD that addresses adaptive rational approximation, or alternatively adaptive TakenakaMalmquist system approximation. The AFD type approximations may be characterized as adaptive approximations by linear combinations of parameterized Szego ̈ and higher order Szego ̈ kernels. This note proposes two kinds of such analytic approximations of which one is called maximalenergy AFDs, including core AFD, Unwending AFD and Cyclic AFD; and the other is again linear combinations of Szego ̈ 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 the imaginary parts of the functions in the sequence approximate, respectively, the original realvalued signals and its Hilbert transform. The two approximations have the same errors in the energy sense due to the fact that Hilbert transformation is an unitary operator in the L2 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.
certainty
"... In this note we will give a survey on adaptive Fourier decompositions in one and multidimensions. The theoretical formulations of three different types of adaptive Fourier decompositions in onedimension, viz., Core AFD, Cyclic AFD in conjunction with best rational approximation and Unwending AFD ..."
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In this note we will give a survey on adaptive Fourier decompositions in one and multidimensions. The theoretical formulations of three different types of adaptive Fourier decompositions in onedimension, viz., Core AFD, Cyclic AFD in conjunction with best rational approximation and Unwending AFD are provided.
Onbackward shift algorithm for estimatingpoles of systems ⋆
"... In this paper, we present an algorithm of estimating poles of linear timeinvariant systems by using the backward shift operator. It is proved that poles of rational functions, including zeros and multiplicities, are solutions of an algebra equation which can be obtained by taking backward shift ope ..."
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In this paper, we present an algorithm of estimating poles of linear timeinvariant systems by using the backward shift operator. It is proved that poles of rational functions, including zeros and multiplicities, are solutions of an algebra equation which can be obtained by taking backward shift operator to normalized reproductive kernels in the unit disc case. The algorithm is accordingly developed for frequencydomain identification. The robustness of this algorithm is proved. Some illustrative examples are presented to show the efficiency for systems with distinguished and multiple poles cases.
MATCHING PURSUITS AMONG SHIFTED CAUCHY KERNELS IN HIGHERDIMENSIONAL SPACES ∗
"... Abstract Appealing to the Clifford analysis and matching pursuits, we study the adaptive decompositions of functions of several variables of finite energy under the dictionaries consisting of shifted Cauchy kernels. This is a realization of matching pursuits among shifted Cauchy kernels in higherd ..."
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Abstract Appealing to the Clifford analysis and matching pursuits, we study the adaptive decompositions of functions of several variables of finite energy under the dictionaries consisting of shifted Cauchy kernels. This is a realization of matching pursuits among shifted Cauchy kernels in higherdimensional spaces. It offers a method to process signals in arbitrary dimensions. Key words Hardy space; monogenic; adaptive decomposition; dictionary; matching pursuit; optimal approximation by rational functions 2010 MR Subject Classification 30G35; 30H10; 41A20; 41A50; 42B30 1
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"... Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis ..."
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Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis