# [time-nuts] Allan variance by sine-wave fitting

Attila Kinali attila at kinali.ch
Mon Nov 27 16:50:00 EST 2017

```On Mon, 27 Nov 2017 19:37:11 +0100
Attila Kinali <attila at 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.

```