On 2014-03-05, Aaron Heller wrote:
The Fourier basis has a countably infinite number of dimensions.
Only if you're talking about the Fourier series or the discrete time Fourier transform. The transform proper is indexed by a continuum.
There are four basic forms of the theory used in signal processing, which are all connected but also subtly different. The Fourier transform is continuous time and continuous frequency. The Fourier series is periodic time and discrete frequency. The discrete time Fourier transform is discrete time and periodic frequency. And finally the discrete Fourier transform is both discrete and periodic in both frequency and in time.
It took me *ages* to get that shit right, and all that goes on between them. I was pretty happy then. Then along came you guys, with your spherical surface harmonics and Fourier-Bessel decompositions. Even the cylindrical variety. A math friend of mine pointed out number theoretical transforms and how this all ties in with locally compact Abelian groups. Abstract harmonical analysis. Then even my engineer pals suddenly went crazy with discrete cosine transforms, MDCT, modulated and lapped transforms, time-frequency decompositions, general partitions of unity, overcomplete bases, L^1 norms, projection pursuits...
Now I drink. -- Sampo Syreeni, aka decoy - de...@iki.fi, http://decoy.iki.fi/front +358-40-3255353, 025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2 _______________________________________________ Sursound mailing list Sursound@music.vt.edu https://mail.music.vt.edu/mailman/listinfo/sursound
