Hi Doug- Regarding this:
"Terms like "well behaived" when applied to the "functon" make me wonder what stipulations might be implied by the language that you'd have to be a formal mathmatician to interpret. As an example, I don't even know what the instrinsic properties of a "function" may be in this context." It turns out to be mostly math details that don't really come up in practice, as somebody (Sampo? Robert?) already mentioned. You have to avoid stuff where the signal blows up to infinity, or has badly-behaved discontinuities or things like that. But in practice, there are no realistic signals that display these issues. The only part where practical signals are problematic is that they are necessarily time-limited, and so cannot be perfectly band-limited. So the Sampling Theorem conditions can't be exactly fulfilled, and we have to live with some (hopefully extremely small) aliasing as a results. But, again, this turns out to be a quite minor issue compared to various of the other practical concerns that come up in designing A/D/A converters. And this one: "If it was just a bunch of random numbers that started somewhere and stopped somewhere, I doubt anyone would be writing equations that mean anything. I'd guess we would turn to statistics at that pint to supply some context." Fourier analysis also works on random signals. But usually in that case we are less interested in the Fourier Transform of the random signals directly, and look more at the Fourier Transform of their correlation functions (this is called the "power spectrum"). That quantity is generally more useful for your usual engineering stuff like filter design, system analysis, etc. If you go to get a graduate degree in signal processing, the core first-year courses are typically what's called "statistical signal processing," which as the name suggests covers signal processing issues in the context of random signals. This is an important, interesting, and worthwhile subject, but also maybe getting a bit afield from the Sampling Theorem issues you are most immediately interested in? E On Thu, Mar 27, 2014 at 11:20 AM, Doug Houghton <[email protected]>wrote: > Some great replies, gives me a lot to think about > > Terms like "well behaived" when applied to the "functon" make me wonder > what > stipulations might be implied by the language that you'd have to be a > formal > mathmatician to interpret. As an example, I don't even know what the > instrinsic properties of a "function" may be in this context. > > Since it's an infinit series I suppose it doesn't really matter, given > enough time you could prove out any rational requirement? which is why you > can throw math at it. If it was just a bunch of random numbers that > started > somewhere and stopped somewhere, I doubt anyone would be writing equations > that mean anything. I'd guess we would turn to statistics at that pint to > supply some context. > > As a broad answer to questions posted in a couple of the replies, my > interest lies in imrpoving my understanding of specifically what the SNST > proves, and the requirements for it to be valid. > > > -- > dupswapdrop -- the music-dsp mailing list and website: > subscription info, FAQ, source code archive, list archive, book reviews, > dsp links > http://music.columbia.edu/cmc/music-dsp > http://music.columbia.edu/mailman/listinfo/music-dsp > -- dupswapdrop -- the music-dsp mailing list and website: subscription info, FAQ, source code archive, list archive, book reviews, dsp links http://music.columbia.edu/cmc/music-dsp http://music.columbia.edu/mailman/listinfo/music-dsp
