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Fractional finite Fourier transform.

Kedar Khare1, Nicholas George

  • 1The Institute of Optics, University of Rochester, Rochester, New York 14627, USA. kedar@optics.rochester.edu

Journal of the Optical Society of America. A, Optics, Image Science, and Vision
|July 21, 2004
PubMed
Summary

A new fractional finite Fourier transform is defined using prolate spheroidal wave functions. This novel transform offers insights into signal processing and inverse problems, with numerical examples demonstrating its capabilities.

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Area of Science:

  • Signal Processing
  • Mathematical Physics
  • Harmonic Analysis

Background:

  • The finite Fourier transform (FFT) is a fundamental tool in signal processing.
  • Existing fractional Fourier transforms (FRFTs) have limitations in certain applications.
  • The need for a robust fractional finite Fourier transform is recognized.

Purpose of the Study:

  • To define a novel fractional version of the finite Fourier transform.
  • To establish its properties and relationship to existing definitions.
  • To demonstrate its application in signal inversion and analysis.

Main Methods:

  • Utilizing prolate spheroidal wave functions of order zero to define the fractional finite Fourier transform.
  • Analyzing the linearity and additivity properties of the new transform.

Related Experiment Videos

  • Investigating the asymptotic behavior and connection to Namias's FRFT definition.

    • Main Results:

    • A new fractional finite Fourier transform is successfully defined.
    • The transform exhibits linearity and additivity in its index.
    • It is shown that the finite Fourier transform can be inverted using finite frequency range information, with sensitivity to noise.

    Conclusions:

    • The proposed fractional finite Fourier transform provides a new mathematical framework.
    • It offers a method for inverting the finite Fourier transform from limited frequency data.
    • Numerical illustrations validate the forward and inverse fractional finite transforms.