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 <[email protected]>: > On Mon, 27 Nov 2017 19:37:11 +0100 > Attila Kinali <[email protected]> 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 -- [email protected] > To unsubscribe, go to https://www.febo.com/cgi-bin/ > mailman/listinfo/time-nuts > and follow the instructions there. > _______________________________________________ time-nuts mailing list -- [email protected] To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
