On 2015-06-19, robert bristow-johnson wrote:
i thought that, because of my misuse of the Dirac delta (from a mathematician's POV, but not from an EE's POV), i didn't think that the "model of sampling as multiplication by a stream of delta functions" was a living organism in the first place. i thought, from the mathematician's POV, we had to get around this by using the Poisson summation formula [...]
In the framework of tempered distributions, all of that follows as well. You can actually do with Dirac deltas what you'd like to do, and what seems natural. Pretty much the only thing you can't do is freely multiply two distributions together, unless they're not just distributions, but functions as well. Convolve if one of the distributions has compact support, or you land within the conventional L_2 theory, or something like that... But otherwise, you can do the funkiest shit.
Nota bene, this is not EE stuff per se. This is heady math stuff, used to formalize what you EEs wanted to do all along. It's the kind of collaboration where us math freaks provide the rubber...and then you EE folks can finally fuck your sister in peace and certainty. ;)
(Sorry, can't help it, been looking at a lot of stand up comedy of late...)
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