On Wed, Jul 2, 2008 at 1:51 PM, Matt Barber brbrofsvl@gmail.com wrote:
Seriously though, I tend to agree with you -- this should explain my unease about searching for every polynomial possibility with a certain number of points. I want to help out as much as I can, but I just don't want to be the one to close a door on an option. I am only qualified to deliver some of the formulae and maybe do some of the programming, but I don't pack the mathematical guns to do the kinds of analytical work Chuck has been doing.
I have a bit of insight on the math of the problem, because I've been working through some examples. And I still don't have an objective idea how to design the right interpolator for the job. Because there's so many possibilities, I think we should employ a few heuristics to guide the design. I think we are working with 3 main types of variations (please suggest more if possible):
- degree of polynomial
- number of points
- setting constraints on derivatives and points
My observations:
- increasing the degree of polynomial allows to increase the rate of
stop-band falloff (e.g 1/w^3 is best possible for a cubic, 1/w^5 is best possible for 5th degree) 2. increasing number of points allows improved derivatives, leading to better high-frequency response
I think we could even turn this around, and specify aspects of the function, like rate of stop-band falloff and location of -3 dB cutoff frequency (which it turns out, is much lower than the Nyquist frequency). That might be the best case for what we can do.
Chuck