Relations between fractional operations and time-frequency distributions, and their applications

The fractional Fourier transform (FRFT) is a useful tool for signal processing. It is the generalization of the Fourier transform. Many fractional operations, such as fractional convolution, fractional correlation, and the fractional Hilbert transform, are defined from it. In fact, the FRFT can be f...

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Bibliographic Details
Published in:IEEE transactions on signal processing 2001-08, Vol.49 (8), p.1638-1655
Main Authors: PEI, Soo-Chang, DING, Jian-Jiun
Format: Article
Language:English
Subjects:
Applied sciences
Beams (radiation)
Convolution
Correlation
Distribution functions
Exact sciences and technology
Filters
Fourier transforms
Helium
Hilbert space
Information, signal and communications theory
Mathematical methods
Pattern recognition
Signal analysis
Signal and communications theory
Signal processing
Signal representation. Spectral analysis
Signal, noise
Studies
Tables
Telecommunications and information theory
Time frequency analysis
Transforms
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Summary:The fractional Fourier transform (FRFT) is a useful tool for signal processing. It is the generalization of the Fourier transform. Many fractional operations, such as fractional convolution, fractional correlation, and the fractional Hilbert transform, are defined from it. In fact, the FRFT can be further generalized into the linear canonical transform (LCT), and we can also use the LCT to define several canonical operations. In this paper, we discuss the relations between the operations described above and some important time-frequency distributions (TFDs), such as the Wigner distribution function (WDF), the ambiguity function (AF), the signal correlation function, and the spectrum correlation function. First, we systematically review the previous works in brief. Then, some new relations are derived and listed in tables. Then, we use these relations to analyze the applications of the FRPT/LCT to fractional/canonical filter design, fractional/canonical Hilbert transform, beam shaping, and then we analyze the phase-amplitude problems of the FRFT/LCT. For phase-amplitude problems, we find, as with the original Fourier transform, that in most cases, the phase is more important than the amplitude for the FRFT/LCT. We also use the WDF to explain why fractional/canonical convolution can be used for space-variant pattern recognition.
ISSN:1053-587X
1941-0476
DOI:10.1109/78.934134