Great to follow this Ross, even with my weak powers of math its informative.
So, just an application note: of course the idea of "cheap" oscillators with interesting band limited waveforms, that require no more than a phasor and arithmetic (multiplies, integer powers, etc) is a goal. I did some experiments with Bezier after being hugely inspired by the sounds Jagannathan Sampath got with his DIN synth. (http://dinisnoise.org/) Jag told me that he had a cute method for matching the endpoints of the segment (you can see in the code), and listening, sounds seem to be alias free, but we never could arrive at a proof of that. Now I am revisiting that territory for another reason and wondering about the properties of easily computed polynomials again. all best, Andy n Sun, Jun 12, 2016 at 02:57:41PM +1000, Ross Bencina wrote: > On 12/06/2016 3:05 AM, Andy Farnell wrote: > >Does it make any sense to talk about the "spectrum of a polynomial" > >over some (periodic) interval (less than infinity)?? Or is that > >silly talk? > > For the infinite interval: > > Expanding the definition of the Fourier transform, for polynomial p: > > P(w) = integral -infinity to infinity p(x) [e^(-2 pi i x w) ] dx > > w is a real number. > > This integral diverges as a Riemann integral. The Cauchy Principle > value for polynomials of strictly odd order is zero. I don't know > whether there's another theory of integration where this Fourier > integral would make sense. > > Looking at transform 308 in the tables here: > https://en.wikipedia.org/wiki/Fourier_transform > > It appears that if you know about distribution theory (I don't) and > the derivatives of the Dirac delta you might be able to make sense > of it. > > But clearly a polynomial over an infinite interval is not going to > make for a very useful signal :) > > ~ > > For a finite (non-periodic) interval, you're essentially talking > about a polynomial windowed with a rectangular window. Such a > function has finite support, so the Fourier integral can be > evaluated over a finite interval. > > Consider an integral related to the Fourier integral for James' > function with a = 2, b = 1, i.e. f(x) = (1-x^2) (the Welch window). > > F(k) = integral -1 to 1 [(1-x^2)e^(ikx)] dx > > Integration by parts (twice) yeilds: > > F(k) = (-4/k^2)cos(k) + (4/k^3)sin(k) > > For higher powers of a and b, you'll end up with a cascade of > roughly ab integration by parts, and the decay of the transform is > something like 1/k^ab. > > ~ > > Now, how best to evaluate the Fourier integral for a repeated > (periodic) polynomial segment? > > Cheers, > > Ross. > > _______________________________________________ > dupswapdrop: music-dsp mailing list > music-dsp@music.columbia.edu > https://lists.columbia.edu/mailman/listinfo/music-dsp >
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