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Some thoughts on 1/f noise.

Now that it's spring here in Norway, the nights are getting shorter and shorter, which on the contrary means that one wakes-up from time to time at early morning 03-04:00 a.m. due to the rising sun and in particular the birds singing around the forest.

One of the mornings last week my sleep was disturbed by a few ravens and crows measuring powers in the very prestigious annual contest "Ugly craw" organized by the local animal union committee, headed by the main moose Mr. Elg. I noticed that while the ravens were taking part in the contest, all other (smaller) types of birds continued to "applaud" all the time while the ravens were fighting. While I was half-asleep the term 1/f noise struck my mind.

Of course it is widely known that many natural phenomena follow a 1/f distribution, or in simple words the higher the power the less frequent the event would be and vice-versa. Half-asleep I did try to think and correlate the birds' songs or the species themselves. In one of the other days I decided to record some of the bird songs and perform a DFT for various recorded samples and then tried to average the f-plots to see whether there is any such 1/f dependency. The results which I show below might not be a full success due to various non-idealities and limited sample sets, but at least they hint an already known fact about 1/f and possibly show that a number of things might go wrong even with a simple time-frequency transformation as the DFT I used.

There are a number of papers about 1/f in human cognition, if interested, I suggest looking at:

"1/f noise in music and speech", Richard Voss, John Clarke, Nature vol. 258, November 27, 1975

"1/f noise in human cognition", D. Gilden, T. Thornton, Science vol. 267, March 24, 1995

"1/f noise a pedagogical review", Eduardo Milotti

To provide a picture of the 1/f occurrence in many "systems" I dare to provide a reprint from "1/f noise" Lawrence M. Ward and Priscilla E Greenwood (2007), Scholarpedia

Examples of 1/f noises occurring in many systems, source:Scholarpedia

This figure is staggering!

Let's start by hearing the bird sample I used.

You might notice that the electronic hum from the microphone preamp of my computer is simply in the same order of magnitude as the birds' songs. This makes these samples very difficult to analyse as we are interested only in the bird content and any other noise (electronic, numerical etc...) should be low. A somewhat higher dynamic range samples would have given a better start.

There is a number of ways to perform this measurement and some might argue, is it the various bird species' songs combined together from a listener's point of hearing that should all be counted for the 1/f measurement, or, is it a single bird, that should be isolated for the measurement? In the current case I possibly isolate a set of birds (bigger species) as I try to filter-out high frequencies and focus my DFT only on the range up-to 60Hz. Here is a block diagram of the signal chain for my analysis:

An overview of the audio signal chain.

After reading the input sample in wav format, a 10th order low-pass butterworth filter with a cutoff frequency of 10kHz is applied. Further after sample squaring the signal is fed through a second filter of the same type with a cutoff of 60Hz. After this extraction (with some non-idealities from the filters) the continuous-time sample information up-to 60Hz is represented in the f-domain by DFT.

Now the question arises, how large window should one have to get accurate enough DFT information for very very low frequencies 0.1Hz - 10Hz? Possibly also another question, how long sample should one have in order to be able to obtain good (with enough oversampling) 1/f plots? We know that the frequency resolution is dependent on the relationship between the input signal sampling rate and the DFT window length. In the current case we have a sampling frequency of 44.1kHz then if we collect 1024 samples (pretty standard number) for the DFT we will have a frequency bin resolution of:

This size is clearly not enough, so as a suggestion to get a 0.1Hz resolution we need about:

We can see that for such low frequencies we need a significant size of the DFT window. This on the other hand would impose that the length of our "bird song" sample must also be quite long to get some meaningful averaged 1/f plots. For instance the 441k samples relate to about 10 seconds of sample time.

Now these birds have a tendency to make quite significant pauses between their squawks sometimes even over a few minutes. If we also need to follow the basic engineering rule of thumb that in order to get any reasonable data the size of the window should be about 5 times the minimum window criteria we get 2.205 Msamples. Taking such a huge sampling window would grately degrade the temporal resolution for the analysis. We still don't care about frequencies beyond 40-60 Hz, but still. For example, if a craw has been squawking 15 times for 25 seconds and later-on during the next 25 seconds (from our sample window time) 20 times, we would still see peaks for both frequencies. The temporal resolution becomes even worse for smaller birds, which squawk even more frequently.

All these simple facts make this 1/f analysis quite subjective with the methods used, strictly speaking, a few minute only samples with low dynamic range and having to trade-off between temporal and frequency resolution. Nevertheless here are some plots of a few samples, not only "bird" content.

Some bird samples.

Some bird 1/f approximations. alpha ~= 1.3

A news emission of the Bulgarian National Radio on 17 May 2014

A news emission 1/f approximations. alpha ~= 1.9

Pink Floyd's Comfortably Numb guitar solo from the 1994 P.U.L.S.E. (The Division Bell tour) concert in Earls Court, London.

Pink Floyd's Comfortably Numb guitar solo 1/f approximations. alpha ~= 1.5

Would you call it spectral leakage, poor temporal/frequency resolution tradeoff, non-integer sampling, poor SNR of the audio samples, filter distortion due to passband ripple, the plots somehow do not look very clean. One is certain, the dependency is 1/f^alpha and alpha measures roughly between ~1.3 for the birds' songs, ~1.9 for the news emission and ~1.5 for the guitar solo.

Even injecting a 50Hz signal to the samples shows signs of some distortion from the ripple in butterworth filters, plus non-integer sampling.

50 Hz sinewave fed through the 10 and 5th order butterworth filters

You can find the octave scripts here.

All aforementioned was/is a nice exercise showing that often applied analyses and measurements require compromises which are only up to the engineer's cognition. There is no very right or sharply wrong approach in this analog world. As for the 1/f, a question arises, is this "law" also applicable to human stereotypes?

Date:Sun May 25 18:56:00 CEST 2014


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