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- Hacker News
- I think 1/f is the uniform measure for scaling rather than translations. It's to multiplication what the standard uniform measure is for addition.
If you want to be boring you could call it the uniform distribution for log frequency.
by shiandow - 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]"
- The comment section at the bottom of the article is pretty interesting. 20 years of people thinking about this.by lars
- 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.
by mturmon - 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 decades old textbooks about it.
I confirm, the low frequency range looks weird on every DFT plot I ever saw, particularly the audio ones. I just assumed it has something to do with ADC and is probably explained on Wikipedia. It's literally one of the last place on Earth when I'd expect to find unsolved mysteries.
by TeMPOraL - 1/f noise basically kills averaging. You collect more signal but at the same time equally more noise.by threatripper
- 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 approach is generating in chunks using Fourier transform from a desired shape. Given the ubiquity of 1/f noise, you'd think there would be a simpler and more direct algorithm, but no.
by cousin_it - 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.
by abetusk