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Wavelet variations on the Shannon sampling theorem

H Bray1, K McCormick, R O Wells

  • 1Dept of Mathematics, Rice University, Houston, TX 77251, USA.

Bio Systems
|January 1, 1995
PubMed
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The Shannon sampling theorem is extended to wavelet expansions. Scaling coefficients uniquely determine square-integrable functions, and a unique variational extension exists for more general functions.

Area of Science:

  • Signal Processing
  • Harmonic Analysis
  • Mathematical Physics

Background:

  • The Shannon sampling theorem is fundamental in signal processing, enabling perfect signal reconstruction from discrete samples.
  • Wavelet analysis offers a powerful alternative to Fourier analysis for representing signals with localized features.

Purpose of the Study:

  • To extend the principles of the Shannon sampling theorem to the domain of wavelet representations.
  • To investigate the unique determination of functions by their wavelet coefficients.
  • To explore variational methods for extending partial wavelet expansions.

Main Methods:

  • Applying the concepts of compactly supported Fourier transforms to wavelet expansions.
  • Analyzing the properties of scaling coefficients in wavelet representations.

Related Experiment Videos

  • Developing a variational approach to minimize translation invariance obstructions.

    • Main Results:

    • Demonstrated that scaling coefficients uniquely determine square-integrable functions under Shannon-like conditions.
    • Established a unique variational extension for more general functions from partial wavelet expansions.
    • Showcased the applicability of sampling theorems in advanced signal processing.

    Conclusions:

    • Wavelet expansions offer a robust framework for function representation and reconstruction, extending classical sampling theory.
    • The developed variational method provides a novel approach for signal completion and analysis.
    • This work bridges classical sampling theory with modern wavelet analysis, opening new avenues in signal processing research.