The Fourier Uncertainty Principle: Not So [Un]Certain?

In a paper titled "Human Time-Frequency Acuity Beats the Fourier Uncertainty Principle" published in Physical Review Letters in January 2013, Jacob N. Oppenheim and Marcelo O. Magnasco present a case for why MP3s suck (that's my capsule summary but it's really much more interesting). From the abstract of their paper:
The time-frequency uncertainty principle states that the product of the temporal and frequency extents of a signal cannot be smaller than 1/(4π). We study human ability to simultaneously judge the frequency and the timing of a sound. Our subjects often exceeded the uncertainty limit, sometimes by more than tenfold, mostly through remarkable timing acuity. Our results establish a lower bound for the nonlinearity and complexity of the algorithms employed by our brains in parsing transient sounds, rule out simple “linear filter” models of early auditory processing, and highlight timing acuity as a central feature in auditory object processing.
In other words, the scientific foundations of lossy codes including MP3s which are based on psycho-acoustic models may in fact be based on incorrect assumptions about our perception.
In many applications such as speech recognition or audio compression (e.g., MP3), the first computational stage consists of generating from the source sound sonogram snippets, which become the input to latter stages. Our data suggest this is not a faithful description of early steps in auditory transduction and processing, which appear to preserve much more accurate information about the timing and phase of sound components than about their intensity.
The test subjects for this research included non-musicians, musicians, composers and conductors and the test results found that these different groups had differing hearing acuity (i.e. some had more Golden Ears than others):
It is important to stress where the difficulty of the task lies. Our preliminary testing included nonmusicians, who where often close in performance to musicians on tasks 1 and 2 (separate time and frequency acuity), but then found tasks 3 and 4 hard (3 is frequency only but with the flanking high note as a distractor, and task 4 is timing only, with the leading note as a distractor), while musicians, trained to play in ensembles, found them easy.

We further found that composers and conductors achieved the best results in task 5 (subjects are asked to discriminate simultaneously whether the test note is higher or lower in frequency than the leading note, and whether the test note appears before or after the flanking high note), consistently beating the uncertainty principle by factors of 2 or more, whereas performers were more likely to beat it only by a few percentage points.

I first learned about this paper from an article in Ars Technica by Chris Lee. Mr Less concludes:
The obvious conclusion, of course, is that humans don't perceive sound linearly. To a large extent, this was already known. We know volume is perceived nonlinearly, but we didn't really know much about temporal/frequency perceptions. Researchers suspected that this was nonlinear—because the brain is anything but linear—but they didn't know which model would accurately represent what goes on in the brain. Researchers and sound engineers have continued to work with linear models because they don't really know what else to use.
Mr. Lee closes with this jab, "I don't have a lot of time for audiophiles with gold-coated connectors and 'unidirectional' coaxial cable, but this data is something I could buy into." But these findings can also point to why different people, especially audiophiles who spend a lot of time and energy just listening, may respond to the same hi-fi gear differently since some people are more in tune with frequency, others more in tune with time/timing, and others still perceive both better than most and different hi-fi gear excels more and less in reproducing these same values. It also helps explain why for some people MP3s are good enough while for others they aren't.

You can purchase "Human Time-Frequency Acuity Beats the Fourier Uncertainty Principle" from Physical Review Letters or download the PDF for free from here.

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judmarc's picture

Thanks for the arxiv link - didn't know the full text had made it onto the arxiv.

Think of some other potential implications of this research.  First, the "sound" of a DAC is determined in significant part by its filtering function(s).  Those functions are constructed using Fourier transforms.  From a series of filters constructed in this way, one or more are chosen on the basis of a compromise between time-domain and frequency-domain performance.  But human hearing performs better than Fourier transforms can in simultaneous time-domain and frequency-domain discrimination.  So the mathematical basis of the sound of virtually every DAC, as well as the basis for instrumented testing and measurement of DAC performance, is less capable than human hearing in important aspects of signal processing.

Fourier analysis (usually as FFTs) is also one of the major tools for design, testing and measurement of other audio components, such as speakers.

Thus one of the major tools for creating and selecting the audio components we purchase is demonstrably less good than our hearing at resolving important aspects of musical signals.

Michael Lavorgna's picture

& very thought-provoking.


Charles Hansen's picture

Dear Michael,

Thanks for bringing this article to the attention of a much broader audioence. "Physical Review" and "Physical Review of Letters" are right at the top when it comes to prestige in the field of peer-reviewed journals in physics.

This stuff makes the "scholarly" article by Meyers and Moran published in the Journal of the Audioi Engineering Society (v.55, n.09) look like the wretched joke that it is. Finally we have some "proof" that the hard-headed nay-sayers will have a very difficult time refuting. (It was only a matter of time.)

Cheers, Charles Hansen, Ayre Acoustics, Inc.

Michael Lavorgna's picture


remlab's picture

Insights like this don't come around very often.

Michael Lavorgna's picture

Provides lots of food for thought.


Baywatcher's picture

Guys, it's "Physical Review Letters" -- no "of." It's the hottest physics journal, and I'm pleased to see a psychophysics article appearing there.

Michael Lavorgna's picture


Gabyb's picture

It'd be interesting to see if any listening tests have been done between digital and analog gear which uses subjects with highest acuity (say composers and conductors per this article) to see if they can hear the quality differences many audiophiles hear, and whether these presumed differences are related to the non-linearity of human hearing.  Also, whether any audio analysis have been done of the playback differences at those sub fourier levels for digital vs analog.

StuLumsden's picture

Well done Michael. Nice catch. This begins to lift a vale. Observation: There is the possibility that all the test subjects "hear" the differences in 1-5 but that only those with training and experience have learned to pay attention to them so as to recognizes them and hence score higher. So, to some extent, almost anyone who chooses to be an audiophile could be. This would say that those for whom MP3 sounds "good enough" are correct. They may have made a conscious choice not to make music/audio as important in their lives. Those who pay attention however expect more and demand it. The same sort of disparity exists between those who choose bulk art over original works of art. Bad art, audio, light – are all small discomforts which can be tolerated to various degrees by various folks. Thankfully, I have never seen anyone react badly to beautiful music faithfully reproduced in a comfortable and artistically pleasing setting.  

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