---------------------------- Original Message ---------------------------- Subject: Re: [music-dsp] a family of simple polynomial windows and waveforms From: "robert bristow-johnson" <r...@audioimagination.com> Date: Sat, June 11, 2016 12:52 am To: music-dsp@music.columbia.edu -------------------------------------------------------------------------- > > ---------------------------- Original Message ---------------------------- > Subject: [music-dsp] a family of simple polynomial windows and waveforms >�From: "James McCartney" <asy...@gmail.com> > Date: Fri, June 10, 2016 9:31 pm > To: music-dsp@music.columbia.edu > -------------------------------------------------------------------------- > >> fun with math: >> >> You can create a family of functions, which can be used as windows, LFO >> waves or envelopes from the formula: >> >> f(x) = (1-x^a)^b >> >> evaluated from x = -1 to +1 >> >> where 'a' is an even positive integer and 'b' is a positive integer. >> >> 'a' controls the flatness of the top and 'b' controls the end tapers. > > some more fun with math: > the integral of f(x) with a=2 gives you an odd-symmetry, odd-order polynomial > that is as linear as it can be at x=0, splices to saturation at |x|=1, and is > continuous in as many > derivatives as possible (i think the number is 2b) at the splice, and has all > derivatives continuous everywhere else (including, of course, the 0th > derivative). > like the smoothest possible soft-clipping.� i guess i forgot to point out the obvious that, as one-shot pulses (or as windows), f(x) splices to a flat zero at x = -1 or +1, with as many (ab-1) continuous derivatives as possible. �integrating that pulse will turn the pulse into a smooth step function with nearly all derivatives continuous everywhere and only a discontinuity in the (ab)th derivative at x = -1 or +1. -- r b-j � � � � � � � � �r...@audioimagination.com "Imagination is more important than knowledge."
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