I'm talking about flicker noise processes 2017-11-27 23:04 GMT+01:00 Mattia Rizzi <mattia.ri...@gmail.com>:
> Hi, > > >To make the point a bit more clear. The above means that noise with > a PSD of the form 1/f^a for a>=1 (ie flicker phase, white frequency > and flicker frequency noise), the noise (aka random variable) is: > 1) Not independently distributed > 2) Not stationary > 3) Not ergodic > > I think you got too much in theory. If you follow striclty the statistics > theory, you get nowhere. > You can't even talk about 1/f PSD, because Fourier doesn't converge over > infinite power signals. > In fact, you are not allowed to take a realization, make several fft and > claim that that's the PSD of the process. But that's what the spectrum > analyzer does, because it's not a multiverse instrument. > Every experimentalist suppose ergodicity on this kind of noise, otherwise > you get nowhere. > > cheers, > Mattia > > 2017-11-27 22:50 GMT+01:00 Attila Kinali <att...@kinali.ch>: > >> On Mon, 27 Nov 2017 19:37:11 +0100 >> Attila Kinali <att...@kinali.ch> wrote: >> >> > X(t): Random variable, Gauss distributed, zero mean, i.i.d (ie PSD = >> const) >> > Y(t): Random variable, Gauss distributed, zero mean, PSD ~ 1/f >> > Two time points: t_0 and t, where t > t_0 >> > >> > Then: >> > >> > E[X(t) | X(t_0)] = 0 >> > E[Y(t) | Y(t_0)] = Y(t_0) >> > >> > Ie. the expectation of X will be zero, no matter whether you know any >> sample >> > of the random variable. But for Y, the expectation is biased to the last >> > sample you have seen, ie it is NOT zero for anything where t>0. >> > A consequence of this is, that if you take a number of samples, the >> average >> > will not approach zero for the limit of the number of samples going to >> infinity. >> > (For details see the theory of fractional Brownian motion, especially >> > the papers by Mandelbrot and his colleagues) >> >> To make the point a bit more clear. The above means that noise with >> a PSD of the form 1/f^a for a>=1 (ie flicker phase, white frequency >> and flicker frequency noise), the noise (aka random variable) is: >> >> 1) Not independently distributed >> 2) Not stationary >> 3) Not ergodic >> >> >> Where 1) means there is a correlation between samples, ie if you know a >> sample, you can predict what the next one will be. 2) means that the >> properties of the random variable change over time. Note this is a >> stronger non-stationary than the cyclostationarity that people in >> signal theory and communication systems often assume, when they go >> for non-stationary system characteristics. And 3) means that >> if you take lots of samples from one random process, you will get a >> different distribution than when you take lots of random processes >> and take one sample each. Ergodicity is often implicitly assumed >> in a lot of analysis, without people being aware of it. It is one >> of the things that a lot of random processes in nature adhere to >> and thus is ingrained in our understanding of the world. But noise >> process in electronics, atomic clocks, fluid dynamics etc are not >> ergodic in general. >> >> As sidenote: >> >> 1) holds true for a > 0 (ie anything but white noise). >> I am not yet sure when stationarity or ergodicity break, but my guess >> would >> be, that both break with a=1 (ie flicker noise). But that's only an >> assumption >> I have come to. I cannot prove or disprove this. >> >> For 1 <= a < 3 (between flicker phase and flicker frequency, including >> flicker >> phase, not including flicker frequency), the increments (ie the difference >> between X(t) and X(t+1)) are stationary. >> >> Attila Kinali >> >> >> -- >> May the bluebird of happiness twiddle your bits. >> >> _______________________________________________ >> time-nuts mailing list -- time-nuts@febo.com >> To unsubscribe, go to https://www.febo.com/cgi-bin/m >> ailman/listinfo/time-nuts >> and follow the instructions there. >> > > _______________________________________________ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
