>all ergodic processes are stationary. (not necessarily the other way around.)
Ah, right, there is no constant mean for a time average to converge to if the process isn't stationary in the first place. Been a while since I worried about the details of ergodicity, mostly I have the intuitive notion that there is no "unreachable" state or infinite memory (ala a fully connected Markov chain). >the reason (besides forgetting stuff i learned 4 decades ago) i left out "stationary" was that i was sorta conflating the two. i just wanted to be able to turn the time-averages in the whatever norm (and L^2 is >as good as any) with probabilistic averages, which is the root meaning of the property "ergodic". but probably "stationary" is a better (stronger) assumption to make. Err, didn't we just establish that ergodicity is the stronger condition? Also I don't think we need to worry about ergodicity in the first place. The process in the OP is ergodic (for P not equal to 0) but we don't need to use that anywhere. We can compute the autocorrelation directly without any reference to time averages or other statistics. We only need ergodicity if we also want to estimate the autocorrelation/psd from example data. Which is important for making plots to verify that the answer is correct, but not needed just to derive the autocorrelation/spectrum themselves. Unless I missed something - where did this ergodicity assumption come from? E On Tue, Nov 10, 2015 at 6:33 PM, robert bristow-johnson < r...@audioimagination.com> wrote: > > > ---------------------------- Original Message ---------------------------- > Subject: Re: [music-dsp] how to derive spectrum of random sample-and-hold > noise? > From: "Ethan Duni" <ethan.d...@gmail.com> > Date: Tue, November 10, 2015 8:58 pm > To: "A discussion list for music-related DSP" < > music-dsp@music.columbia.edu> > -------------------------------------------------------------------------- > > >>(Semi-)stationarity, I'd say. Ergodicity is a weaker condition, true, > >>but it doesn't then really capture how your usual L^2 correlative > >>measures truly work. > > > > I think we need both conditions, no? > > all ergodic processes are stationary. (not necessarily the other way > around.) > > > > the reason (besides forgetting stuff i learned 4 decades ago) i left out > "stationary" was that i was sorta conflating the two. i just wanted to be > able to turn the time-averages in the whatever norm (and L^2 is as good as > any) with probabilistic averages, which is the root meaning of the property > "ergodic". but probably "stationary" is a better (stronger) assumption to > make. > > > > > > > >>Something like that, yes, except that you have to factor in aliasing. > > > > What aliasing? Isn't this process generated directly in the discrete time > > domain? > > i'm thinking the same thing. it's a discrete-time Markov process. just > model it and analyze it as such. assuming stationarity, we should be able > to derive an autocorrelation function (and i think you guys did) and from > that (and the DTFT) you have the (periodic) power spectrum. > > worry about frequency aliasing when you decide to output this to a DAC. > > > > -- > > > > > 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 >
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