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.
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© 2002 Springer Science+Business Media New York
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Brémaud, P. (2002). Fourier Transforms of Finite Energy Signals. In: Mathematical Principles of Signal Processing. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3669-4_10
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DOI: https://doi.org/10.1007/978-1-4757-3669-4_10
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-2956-3
Online ISBN: 978-1-4757-3669-4
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