On Mon, Jul 13, 2015 at 8:39 AM, Charles Z Henry <czhe...@gmail.com> wrote: > On Mon, Jul 13, 2015 at 3:28 AM, Vadim Zavalishin > <vadim.zavalis...@native-instruments.de> wrote: >> On 10-Jul-15 19:50, Charles Z Henry wrote: >>> >>> The more general conjecture for the math heads : >>> If u is the solution of a differential equation with forcing function g >>> and y = conv(u, v) >>> Then, y is the solution of the same differential equation with forcing >>> function >>> h=conv(g,v) >>> >>> I haven't got a solid proof for that yet, but it looks pretty easy. >> >> >> How about the equation >> >> u''=-w*u+g >> >> where v is sinc and w is above the sampling frequency? > > I think you meant sqrt(w) is above the sampling freq. > > This is also a good one. I also saw some scenarios like this, that > just might take a little time. The math should come out that if > sqrt(w) > f_s, u=0 and if sqrt(w) < f_s, u is a sine. I'll work on it > a bit over the week and see if I can make the calculus work :)
--From here on, let's replace w with w^2, to make things simpler in discussion-- My first observations on the general problem: For numerical integration, we need to re-write as a system of 1st-order ODE's in the first place: d/dt u' + w^2*u = g d/dt u - u' = 0 Integrate the 1st equation to get u', and integrate the 2nd equation to get u. I think the general problem (arbitrary order linear ODE's and forcing functions) is better cast as a system of 1st order ODE's. There will be an inductive step, so that increasing the order of the ODE (aka size of the system of 1st order ODE's) does not present a novel problem. -- 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
