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An Interesting Fourier Transform – 1/f Noise (2007)

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Re: An Interesting Fourier Transform – 1/f Noise (2007)

#21

Post by jhallenworld »

TeMPOraL wrote: post #4

Just as interesting and surprising to me is that this 0-100Hz line is unexplained . Given that signal processing is the foundation of pretty much all digital technology, as well as the analog technologies that came before, I've kind of assumed every segment of a frequency plot be well studied and understood, with half a dozen of names to choose for (doublesine quefrency this, Kowalski-Shannon that...) and a heap of d…

1/f noise means that if you wait long enough an asteroid will hit the earth or the sun will go nova, etc.

Surely there is some connection to entropy.

Re: An Interesting Fourier Transform – 1/f Noise (2007)

#22

Post by mturmon »

rcxdude wrote: post #13
threatripper wrote: post #11

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

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...)

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.

Re: An Interesting Fourier Transform – 1/f Noise (2007)

#24

Post by peter_d_sherman »

My favorite part of the article:

>"Here is something even more interesting. As you approach α = -1, the time domain approaches a shape of t-1, and the frequency domain approaches a flat magnitude with a zero phase. However, a flat magnitude and zero phase corresponds to a delta function, δ(t), in the time domain."

Related:

https://en.wikipedia.org/wiki/Dirac_delta_function

>"Indeed, Heaviside introduced the δ-function in his work on electromagnetism and electrical engineering.[14] In a 1963 interview, Dirac stated, "All electrical engineers are familiar with the idea of a pulse, and the δ-function is just a way of expressing a pulse mathematically."[15]"

Re: An Interesting Fourier Transform – 1/f Noise (2007)

#25

Post by [deleted] »

cousin_it wrote: post #17

When I was learning DSP, I was surprised by the fact that generating pink (1/f) noise, sample by sample, is not mathematically easy at all. One practical approach is passing white noise (every sample independent random) through a "pinking filter", which is a sum of a bunch of lowpass filters at different frequencies, so that the sum of their cutoff "knees" approximates the frequency curve of pink noise. Another appro…

[deleted]

Re: An Interesting Fourier Transform – 1/f Noise (2007)

#26

Post by stagger87 »

cousin_it wrote: post #17

When I was learning DSP, I was surprised by the fact that generating pink (1/f) noise, sample by sample, is not mathematically easy at all. One practical approach is passing white noise (every sample independent random) through a "pinking filter", which is a sum of a bunch of lowpass filters at different frequencies, so that the sum of their cutoff "knees" approximates the frequency curve of pink noise. Another appro…

If you change your Fourier transform approach into a time domain convolution, then you can generate it sample by sample. White noise -> FIR filter, this is a pretty simple and direct algorithm as far as DSP goes. The wiki on pink noise mentions this. I've done the FIR filter approach to simulate 1/f phase noise.

Re: An Interesting Fourier Transform – 1/f Noise (2007)

#27

Post by abetusk »

Power law distributions are specializations of the more general Levy stable distributions [0] [1]. Levy stable distributions answer the following question:

Given that the sum of independent and identically distributed random variables that converge to a distribution, what is the distribution they converge to?

If you answered Gaussian, you'd be wrong. The correct answer is Levy stable. There was no condition on finite variance. When variance can be infinite, Levy stable, or power law tail distributions, is the result. When the variance is finite, a Gaussian is the limiting distribution and, consequently, a Gaussian distribution is part of the family of Levy stable distributions.

The stability quality is the reason why the Levy stable (aka power law tail) distributions show up all over the place. If you've ever heard that the reason why the Normal distribution is called "normal", because the sums of (finite variance) random variables converges to a Gaussian, the same reasoning applies to the Levy stable. In some sense, Levy stable distributions are more normal than the normal distribution. My opinion is that infinite variance is hard for people to wrap their heads around, so they reject the premise.

Unfortunately I don't have a good answer for what the article brings up about the Fourier transform, but I'm almost positive that this can be answered with Levy stable distributions in mind. I will say that the distribution is often characterized by it's characteristic function. A short perusal of Wikipedia talks about Levy stable distributions being closed under Fourier transforms, which is what the article is talking about.

[0] https://en.wikipedia.org/wiki/L%C3%A9vy_distribution

[1] https://en.wikipedia.org/wiki/Stable_distribution

Re: An Interesting Fourier Transform – 1/f Noise (2007)

#29

Post by mturmon »

abetusk wrote: post #27

Power law distributions are specializations of the more general Levy stable distributions [0] [1]. Levy stable distributions answer the following question: Given that the sum of independent and identically distributed random variables that converge to a distribution, what is the distribution they converge to? If you answered Gaussian, you'd be wrong. The correct answer is Levy stable. There was no condition on finite…

This is mistaking two kinds of power laws.

OP is considering power laws in the frequency domain.

You are considering power laws in the heavy tail of a distribution.

Different things! But there are confounders that make discussion seem similar:

- questions about moments and convergence (OP: do we have finite energy in the Fourier domain; your comment: do the tails of the distribution fall off fast enough to have finite moments of order 1 or 2)

- questions about averaging (OP: in the time domain; your comment: as an expectation obtained by integrating a distribution, or as a closure property of the stable class of distributions)

Re: An Interesting Fourier Transform – 1/f Noise (2007)

#30

Post by mturmon »

fch42 wrote: post #3
mturmon wrote: post #2

The piece ends with the observation that maybe the fact that 1/f noise is its own Fourier transform is a clue. Turns out this property is not unusual. There are many such pairs - there’s a reasonably well-known journal paper with a construction technique.

The paper linked at the "see also" section ? (thx)

See: https://smg.quora.com/Are-there-any-functions-whose-Fourier-...

Look under the section “Surprise”. This gives a construction that produces a new function g, whose transform is itself, that is a simple modification of any function ”f” that you supply.

In essence, the class of “functions that transform into themselves” is surprisingly broad and non-specific for those of us who key in on that fact about the Gaussian.

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