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Discrete Quadratic-Phase Fourier Transform: Theory and Convolution Structures
Hari M Srivastava1,2,3,4, Waseem Z Lone5, Firdous A Shah5
1Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada.
Entropy (Basel, Switzerland)
|July 8, 2023
Summary
This study introduces the discrete quadratic-phase Fourier transform, a generalized tool for digital signal processing. It expands upon existing transforms, offering new methods for analyzing signal spectra and structures.
Area of Science:
- Digital Signal Processing
- Mathematical Physics
Background:
- The discrete Fourier transform (DFT) is fundamental for analyzing finite-duration signals.
- Existing generalized DFTs like the fractional and linear canonical transforms offer specialized analyses.
Purpose of the Study:
- Introduce the discrete quadratic-phase Fourier transform (DQPFT).
- Unify and generalize various discrete Fourier transform variants.
- Explore fundamental properties and applications of the DQPFT.
Main Methods:
- Formulation of Parseval's and reconstruction formulae for DQPFT.
- Development of weighted and non-weighted convolution structures.
- Establishment of correlation structures within the DQPFT framework.
Main Results:
- The DQPFT is defined and its fundamental properties are established.
- Parseval's and reconstruction formulae are derived.
- Novel convolution and correlation structures are presented for the DQPFT.
Conclusions:
- The DQPFT provides a unified framework for a broader class of discrete transforms.
- This generalization enhances the analytical capabilities in digital signal processing.
- The established convolution and correlation structures offer new tools for signal analysis.
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