On 2014-03-09, Fons Adriaensen wrote:
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.
There are just two, the FT and the DFT. The only difference between
the last three forms you mention is only a matter of interpretation.
You can easily interpret even the FT into the whole. All it takes is
topological completion, and then working with suitable equispaced delta
distributions. Discrete time Fourier transform drops off very naturally
from there and vice versa, you can recover a dense basis essentially
equivalent to the full FT one simply by passing the period of the DTFT
to the null limit. No wiggle-room, nothing.
But of course that wasn't what I was talking about. In a certain sense
they're all the same, which is why I said already that they're
intricately connected. In the sense I was talking about, which is the
more trivial kind, they're nothing of the sort. They really can be
separated by the kind of systematic I laid out in talking about
periodicity and discreteness, and that's pretty much what governs their
actual usage in the mathematical and engineering disciplines. I also
think that way to looking at the Fourier methods is rather useful as
such, *because* of the practicality of the viewpoint.
--
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
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