---------------------------- Original Message ---------------------------- Subject: Re: [music-dsp] a family of simple polynomial windows and waveforms From: "Ross Bencina" <rossb-li...@audiomulch.com> Date: Sat, June 11, 2016 8:24 am To: music-dsp@music.columbia.edu -------------------------------------------------------------------------- > > On 11/06/2016 9:16 PM, Andy Farnell wrote: >> Is there something general for the spectrum of all polynomials? > > I think Robert was referring to the waveshaping spectrum with a > sinusoidal input. yeah, but this thing from James is more specific (bandlimited smooth pulses with *no* Gibbs ringing). �perhaos tge "something general for the spectrum of all polynomials" would be the sum of Tchebyshev polynomials. �with Tchebyshevs driven by a sinusoid (with amplitude of 1), you can straight-forwardly tune the amplitudes of each and every harmonic. > > If the input is a (complex) sinusoid it follows from the index laws: > > (e^(iw))^2 = e^(i2w) > > In excruciating detail*: > > Consider the expansion of (1-z)^b (use the binomial theorem). The > highest power of z in the expansion will be z^b. > > e.g. for b = 3: > > (1-z)^3 = -z^3 + 3z^2 - 3z + 1 > > Similarly, for f(z) = (1-z^a)^b, the highest power of z will be ab. (Not > sure where Robert got a|b| from though). > i was referring to the index of the highest harmonic with non-zero amplitude. � i guess i coulda said |ab| but "a" must be positive. oh, i had brain-fart. �i was thinking that "b" could be negative. �(not the case.) � so it *is* just "ab". sorry to stink up the place. -- r b-j � � � � � � � � �r...@audioimagination.com "Imagination is more important than knowledge."
_______________________________________________ dupswapdrop: music-dsp mailing list music-dsp@music.columbia.edu https://lists.columbia.edu/mailman/listinfo/music-dsp
