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

[DrummingAtom]

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?

 

[marcusl]

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.