Rotations from Fourier Transforms? 4 times gets you back to original.

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SUMMARY

The discussion centers on the concept of Fourier transforms (FT) as rotations in infinite-dimensional space, specifically highlighting that performing the transform four times returns to the original state. The fractional Fourier transform (FrFFT) is introduced as a generalization that allows for arbitrary angle rotations, which is particularly useful for analyzing time-varying signals. Additionally, the Wigner-Ville distribution is mentioned as a related concept that provides further insights into signal processing. This foundational understanding is crucial for advanced studies in signal analysis and processing.

PREREQUISITES
  • Understanding of Fourier transforms and their properties
  • Familiarity with fractional Fourier transforms (FrFFT)
  • Basic knowledge of signal processing techniques
  • Concept of Wigner-Ville distribution
NEXT STEPS
  • Research the mathematical foundations of fractional Fourier transforms (FrFFT)
  • Explore applications of the Wigner-Ville distribution in signal analysis
  • Study the implications of rotations in infinite-dimensional spaces
  • Investigate time-varying signal processing techniques
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Students in electrical engineering, signal processing professionals, and anyone interested in advanced mathematical concepts related to Fourier transforms and their applications in analyzing signals.

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Today in my circuits class, we were talking about Fourier transforms and my professor briefly said something about how a Fourier transform is a rotation in infinite dimensional space. I would ask him more about it but since it's beyond our course I'd rather not bug him. Where can I learn more about this concept? Or what exactly is going on with that idea?
 
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The FT can be looked at as a "90° transform" from the space axis to the time axis. (I'm not sure about the infinite dimensional part, unless it refers to the infinitude of points in a continuous waveform.) Going around four times returns you to the starting point, as your professor noted. Do a search for the fractional Fourier transform, which is a rotation through arbitrary angles, to see its generalization. The FrFFT can be useful when looking at time-varying signals. The older Wigner-Ville distribution is related, as well.
 

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