Discrete linear canonical transform based on hyperdifferential operators

Date
2019-05
Advisor
Instructor
Source Title
IEEE Transactions on Signal Processing
Print ISSN
1053-587X
Electronic ISSN
1941-0476
Publisher
IEEE
Volume
67
Issue
9
Pages
2237 - 2248
Language
English
Type
Article
Journal Title
Journal ISSN
Volume Title
Abstract

Linear canonical transforms (LCTs) are of importance in many areas of science and engineering with many applications. Therefore, a satisfactory discrete implementation is of considerable interest. Although there are methods that link the samples of the input signal to the samples of the linear canonical transformed output signal, no widely-accepted definition of the discrete LCT has been established. We introduce a new approach to defining the discrete linear canonical transform (DLCT) by employing operator theory. Operators are abstract entities that can have both continuous and discrete concrete manifestations. Generating the continuous and discrete manifestations of LCTs from the same abstract operator framework allows us to define the continuous and discrete transforms in a structurally analogous manner. By utilizing hyperdifferential operators, we obtain a DLCT matrix, which is totally compatible with the theory of the discrete Fourier transform (DFT) and its dual and circulant structure, which makes further analytical manipulations and progress possible. The proposed DLCT is to the continuous LCT, what the DFT is to the continuous Fourier transform. The DLCT of the signal is obtained simply by multiplying the vector holding the samples of the input signal by the DLCT matrix.

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Other identifiers
Book Title
Keywords
Linear canonical transform (LCT), Fractional Fourier transform (FRT), Operator theory, Discrete transforms, Hyperdifferential operators
Citation
Published Version (Please cite this version)