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Fourier Transforms of Finite Energy Signals

In: Mathematical Principles of Signal Processing

Author

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  • Pierre Brémaud

    (École Polytechnique Fédérale de Lausanne
    INRIA/École Normale Supérieure)

Abstract

A stable signal as simple as the rectangular pulse has a Fourier transform that is not integrable, and therefore one cannot use the Fourier inversion theorem for stable signals as it is. However, there is a version of this inversion formula that applies to all finite-energy functions (for instance, the rectangular pulse). The analysis becomes slightly more involved, and we will have to use the framework of Hilbert spaces. This is largely compensated by the formal beauty of the results, due to the fact that a square-integrable function and its FT play symmetrical roles.

Suggested Citation

  • Pierre Brémaud, 2002. "Fourier Transforms of Finite Energy Signals," Springer Books, in: Mathematical Principles of Signal Processing, chapter 0, pages 155-160, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4757-3669-4_10
    DOI: 10.1007/978-1-4757-3669-4_10
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