An Interesting Fourier Transform – 1/f Noise (2007)
dsprelated.com
dsprelated.com
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.
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)
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.
(though of course, a random walk means you'll get even worse as you measure for longer...)
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.
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...
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.
(Also, the low frequency range on a DFT can look weird for reasons other than noise: it'll also tend to rise up if there's any longer-term structure to the signal as well, so you need to be careful interpreting them blindly if you're trying to measure noise)
So its existence is often largely treated as an empirical rule of thumb rather than having a specific physical cause.
The other thing to note is that the noise plot in a dataset is probably a curve fit.
Surely there is some connection to entropy.
Turns out this property is not unusual. There are many such pairs - there’s a reasonably well-known journal paper with a construction technique.
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.
(The frequency is 42.3338481 nanohertz.)
>"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]"
If you want to be boring you could call it the uniform distribution for log frequency.
Actual white noise indeed sounds really bad and grating. The best for me was a mix of pink and brown noise, pink for an ~equal baseline and brown to make it sound a little more mellow.
I suspect most/all generators meant to block out noise do something similar. It really sounds pretty bad without that, especially for anything more than a few seconds.