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**1 - 7**of**7**### Frequency*Distribution*Based*on*Mono-component*Decompositions Transient Time-Frequency Distribution Based on Mono-component Decompositions

, 2012

"... In this paper we propose a new type of non-negative time-frequency distribution asso-ciated with mono-components in both the non-periodic and periodic cases, called tran-sient time-frequency distribution (TTFD), and study its properties. The TTFD of a mono-component signal can be obtained directly t ..."

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In this paper we propose a new type of non-negative time-frequency distribution asso-ciated with mono-components in both the non-periodic and periodic cases, called tran-sient time-frequency distribution (TTFD), and study its properties. The TTFD of a mono-component signal can be obtained directly through its analytic instantaneous frequency. The characteristic property of TTFD is its complete concentration along the analytic instantaneous frequency graph. For multi-components there are induced time-frequency distributions called composing transient time-frequency distribution (CT-TFD). Each CTTFD is defined as the superposition of the TTFDs of the composing intrinsic mono-components in a suitable mono-components decomposition of the tar-geted multi-component. In studying the properties of TTFD and CTTFD the relations between the Fourier frequency and analytic instantaneous frequency are examined.

### 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 varia-tions 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 varia-tions 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.

### certainty

"... In this note we will give a survey on adaptive Fourier decompositions in one- and multi-dimensions. The theoretical formulations of three different types of adaptive Fourier decompositions in one-dimension, 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 multi-dimensions. The theoretical formulations of three different types of adaptive Fourier decompositions in one-dimension, viz., Core AFD, Cyclic AFD in conjunction with best rational approximation and Unwending AFD are provided.

### MATCHING PURSUITS AMONG SHIFTED CAUCHY KERNELS IN HIGHER-DIMENSIONAL 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 con-sisting of shifted Cauchy kernels. This is a realization of matching pursuits among shifted Cauchy kernels in higher-d ..."

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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 con-sisting of shifted Cauchy kernels. This is a realization of matching pursuits among shifted Cauchy kernels in higher-dimensional spaces. It offers a method to process signals in arbitrary dimensions. Key words Hardy space; monogenic; adaptive decomposition; dictionary; matching pur-suit; optimal approximation by rational functions 2010 MR Subject Classification 30G35; 30H10; 41A20; 41A50; 42B30 1

### Received (Day Month Year)

, 2013

"... Communicated by (xxxxxxxxxx) A sequence of special functions in Hardy space H2(Ts) are constructed from Cauchy kernel on unit disk D. Applying projection operator of the sequence of functions leads to an analytic sampling approximation to f, any given function in H2(Ts). That is, f can be approximat ..."

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Communicated by (xxxxxxxxxx) A sequence of special functions in Hardy space H2(Ts) are constructed from Cauchy kernel on unit disk D. Applying projection operator of the sequence of functions leads to an analytic sampling approximation to f, any given function in H2(Ts). That is, f can be approximated by its analytic samples in Ds. Under a mild condition, f is approxi-mated exponentially by its analytic samples. By the analytic sampling approximation, a signal in H2(T) can be approximately decomposed into components of positive instanta-neous frequency. Using circular Hilbert transform, we apply the approximation scheme in Hs(Ts) to Ls(T2) such that a signal in Ls(T2) can be approximated by its analytic samples on Cs. A numerical experiment is carried out to illustrate our results.

### unknown title

"... 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