• Title, Summary, Keyword: Fourier transform

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FOURIER-FEYNMAN TRANSFORM AND CONVOLUTION OF FOURIER-TYPE FUNCTIONALS ON WIENER SPACE

  • Kim, Byoung Soo
    • East Asian mathematical journal
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    • v.29 no.5
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    • pp.467-479
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    • 2013
  • We develop a Fourier-Feynman theory for Fourier-type functionals ${\Delta}^kF$ and $\widehat{{\Delta}^kF}$ on Wiener space. We show that Fourier-Feynman transform and convolution of Fourier-type functionals exist. We also show that the Fourier-Feynman transform of the convolution product of Fourier-type functionals is a product of Fourier-Feynman transforms of each functionals.

ON UNIFORM SAMPLING IN SHIFT-INVARIANT SPACES ASSOCIATED WITH THE FRACTIONAL FOURIER TRANSFORM DOMAIN

  • Kang, Sinuk
    • Honam Mathematical Journal
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    • v.38 no.3
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    • pp.613-623
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    • 2016
  • As a generalization of the Fourier transform, the fractional Fourier transform plays an important role both in theory and in applications of signal processing. We present a new approach to reach a uniform sampling theorem in the shift-invariant spaces associated with the fractional Fourier transform domain.

QUANTUM EXTENSIONS OF FOURIER-GAUSS AND FOURIER-MEHLER TRANSFORMS

  • Ji, Un-Cig
    • Journal of the Korean Mathematical Society
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    • v.45 no.6
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    • pp.1785-1801
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    • 2008
  • Noncommutative extensions of the Gross and Beltrami Laplacians, called the quantum Gross Laplacian and the quantum Beltrami Laplacian, resp., are introduced and their basic properties are studied. As noncommutative extensions of the Fourier-Gauss and Fourier-Mehler transforms, we introduce the quantum Fourier-Gauss and quantum Fourier- Mehler transforms. The infinitesimal generators of all differentiable one parameter groups induced by the quantum Fourier-Gauss transform are linear combinations of the quantum Gross Laplacian and quantum Beltrami Laplacian. A characterization of the quantum Fourier-Mehler transform is studied.

CONDITIONAL FOURIER-FEYNMAN TRANSFORMS OF VARIATIONS OVER WIENER PATHS IN ABSTRACT WIENER SPACE

  • Cho, Dong-Hyun
    • Journal of the Korean Mathematical Society
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    • v.43 no.5
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    • pp.967-990
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    • 2006
  • In this paper, we evaluate first variations, conditional first variations and conditional Fourier-Feynman transforms of cylinder type functions over Wiener paths in abstract Wiener space and then, investigate relationships among first variation, conditional first variation, Fourier-Feynman transform and conditional Fourier-Feynman transform of those functions. Finally, we derive the conditional Fourier-Feynman transform for the product of cylinder type function which defines the functions in a Banach algebra introduced by Yoo, with n linear factors.

Phase Retrieval Using an Additive Reference Signal: I. Theory (더해지는 기준신호를 이용한 위성복원: I. 이론)

  • Woo Shik Kim
    • Journal of the Korean Institute of Telematics and Electronics B
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    • v.31B no.5
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    • pp.26-33
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    • 1994
  • Phase retrieval is concerned with the reconstruction of a signal from its Fourier transform magnitude (or intensity), which arises in many areas such as X-ray crystallography, optics, astronomy, or digital signal processing. In such areas, the Fourier transform phase of the desired signal is lost while measuring Fourier transform magnitude (F.T.M.). However, if a reference 'signal is added to the desired signal, then, in the Fourier trans form magnitude of the added signal, the Fourier transform phase of the desired signal is encoded. This paper addresses uniqueness and retrieval of the encoded Fourier phase of a multidimensional signal from the Fourier transform magnitude of the added signal along with the Fourier transform magnitude of the desired signal and the information of the additive reference signal. In Part I, several conditions under which the desired signal can be uniquely specified from the two Fourier transform magnitudes and the additive reference signal are presented. In Part II, the development of non-iterative algorithms and an iterative algorithm that may be used to reconstruct the desired signal(s) is considered.

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Phase Retrieval Using an Additive Reference Signal: II. Reconstruction (더해지는 기준신호를 이용한 위성복원: II. 복원)

  • Woo Shik Kim
    • Journal of the Korean Institute of Telematics and Electronics B
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    • v.31B no.5
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    • pp.34-41
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    • 1994
  • Phase retrieval is concerned with the reconstruction of a signal from its Fourier transform magnitude (or intensity), which arises in many areas such as X-ray crystallography, optics, astronomy, or digital signal processing In such areas, the Fourier transform phase of the desired signal is lost while measuring Fourier transform magnitude (F.T.M.). However, if a reference 'signal is added to the desired signal, then, in the Fourier trans form magnitude of the added signal, the Fourier transform phase of the desired signal is encoded This paper addresses uniqueness and retrieval of the encoded Fourier phase of a multidimensional signal from the Fourier transform magnitude of the added signal along with Fourier transform magnitude of the desired signal and the information of the additive reference signal In Part I, several conditions under which the desired signal can be uniquely specified from the two Fourier transform magnitudes and the additive reference signal are presented In Part II, the development of non-iterative algorithms and an iterative algorithm that may be used to reconstruct the desired signal (s) is considered

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LFM Signal Separation Using Fractional Fourier Transform (Fractional Fourier 변환을 이용한 LFM 신호 분리)

  • Seok, Jongwon;Kim, Taehwan;Bae, Keunsung
    • Journal of the Korea Institute of Information and Communication Engineering
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    • v.17 no.3
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    • pp.540-545
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    • 2013
  • The Fractional Fourier transform, as a generalization of the classical Fourier Transform, was first introduced in quantum mechanics. Because of its simple and useful properties of Fractional Fourier transform in time-frequency plane, various research results in sonar and radar signal processing have been introduced and shown superior results to conventional method utilizing Fourier transform until now. In this paper, we applied Fractional Fourier transform to sonar signal processing to detect and separate the overlapping linear frequency modulated signals. Experimental results show that received overlapping LFM(Linear Frequency Modulation) signals can be detected and separated effectively in Fractional Fourier transform domain.

FOURIER-TYPE FUNCTIONALS ON WIENER SPACE

  • Chung, Hyun-Soo;Tuan, Vu Kim
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.3
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    • pp.609-619
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    • 2012
  • In this paper we define the Fourier-type functionals via the Fourier transform on Wiener space. We investigate some properties of the Fourier-type functionals. Finally, we establish integral transform of the Fourier-type functionals which also can be expressed by other Fourier-type functionals.

A Comparison between Abel-Fourier and Digital Linear Filter Methods (Abel-Fourier법과 디지탈 선형필터법과의 비교)

  • Kim, Hee Joon
    • Economic and Environmental Geology
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    • v.20 no.2
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    • pp.119-123
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    • 1987
  • Hankel transform of order 0 can be represented by a combination of Abel transform and Fourier transform. This Abel-Fourier method can offer computational advantages over a conventional digital linear filter method through the uses of a rapid Abel transform with shift variant recursive filter and of a fast Fourier transform. The Abel-Fourier method, however, is generally less accurate than a well-designed digital linear filter method. In geoelectrical applications, the digital linear filter method seems to be more flexible than the Abel-Fourier method.

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