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Comment by threatripper | Hacker News Reader

1/f noise basically kills averaging. You collect more signal but at the same time equally more noise.

1/f noise does not truly kill averaging, observe how for increasing f, the noise spectrum looks "white" locally, but with a decreasing noise power for ever higher frequencies.

the 1/f noise at significant power levels is restricted to the lower and lower frequencies, effectively a slowly varying reference ("0") voltage of the amplifier, measuring ADC ground every other sample effectively recalibrates the offset voltage of the amplifier, think of "correlated double sampling".

Effectively measuring in sequence 0V, signal, 0V, signal, ... moves the signal of interest to a higher frequency band, where the 1/f noise is more tame.

When a cliff blocks the way of a vehicle, we don't say "cliffs kill vehicle travel", insteas we just drive around it...

Yup, it has the fun property that you'll get the same error in your estimate of the mean regardless of averaging time (though since it's not stationary this isn't even really that well defined).

(though of course, a random walk means you'll get even worse as you measure for longer...)

This makes me think of the Cauchy distribution, a probability distribution whose average follows the distribution itself rather than converging (hence, the distribution has no "mean" despite being symmetric). Is there any connection here, or is that just a coincidental similarity?

At the same time, though, it doesn’t mean you can’t infer the location parameter of a Cauchy distribution with more and more precision as you receive more samples. It just means that averaging the samples is not the way to do it.

Yes. My PhD advisor had a research interest in axiom systems for probability that are weaker than the familiar Kolmogorov axioms, which are sometimes abbreviated "CMP" for "conventional mathematical probability".

I'll try to remember the setup. The CMP axioms imply that, in a shift-invariant system X(t) (which is a different class than "stationary" -- not necessarily implying existence of second moments), if the mean of X(t) exists finite, then a long-term average of X(t) must converge.

However, you can observe time-invariant physical systems (such as a noisy resistor in a static environment) with spectra that obey the 1/f law down to very low frequencies (i.e., over very long time baselines) -- the time average does not converge. My advisor had a stack of magnetic tapes on his bookshelf with such samples.

These systems would seem to be disobeying the axioms of CMP, thereby motivating searches for alternative formulations that are more general.

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