---------------------------- Original Message ---------------------------- Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold noise? From: "Ross Bencina" <rossb-li...@audiomulch.com> Date: Wed, November 4, 2015 12:22 am To: r...@audioimagination.com music-dsp@music.columbia.edu -------------------------------------------------------------------------- > On 4/11/2015 9:39 AM, robert bristow-johnson wrote: >> i have to confess that this is hard and i don't have a concrete solution >> for you. > > Knowing that this isn't well known helps. I have an idea (see below). It > might be wrong. > > >> it seems to me that, by this description: >> >> r[n] = uniform_random(0, 1) >> >> if (r[n] <= P) >> >> x[n] = uniform_random(-1, 1); >> >> else >> >> x[n] = x[n-1]; >> >> from that, and from the assumption of ergodicity (where all time >> averages can be replaced with probabilistic averages), then it should be >> possible to derive an autocorrelation function from this. >> >> but i haven't done it. > > Using AMDF instead of autocorrelation: > > let n be an arbitrary time index > let t be the AMDF lag time of interest > > AMDF[t] = fabs(x[n] - x[n-t]) > > there are two cases: > > case 1, (holding): x[n-t] == x[n] > case 2, (not holding) x[n-t] == uniform_random(-1, 1) > > In case 1, AMDF[t] = 0 > In case 2, AMDF[t] = 2/3 (i think?) � so if it's ASDF, it might be 1/3 > > To get the limit of AMDF[t], weight the values of the two cases by the > probability of each case case. (Which seems like a textbook waiting-time > problem, but will require me to return to my textbook). > > Then I just need to convert the AMDF to PSD somehow. � it *is* possible to convert from ASDF to autocorrelation (and then to PSD). �square the difference instead of fabs(). > > Does that seem like a reasonable approach? > perhaps. -- � r b-j � � � � � � � � � r...@audioimagination.com � "Imagination is more important than knowledge."
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