Human Hearing Acuity Shown to be More Accurate Than Standard Linear Models

While it is well known that human perception of sound wave amplitude, volume, is nonlinear, taking a logarithmic form, it has long been assumed that frequency and temporal information are perceived in a linear fashion.  In a paper published in Physical Review Letters earlier this year, researchers Jacob Oppenheim and Marcelo Magnasco from the Laboratory of Mathematical Physics at Rockefeller University have shown that the commonly held assumption that early stages of auditory processing in human hearing behave in a linear fashion is likely incorrect and actually is nonlinear as well.  The paper, titled Human Time-Frequency Acuity Beats the Fourier Uncertainty Principle describes the first direct psychoacoustical testing of the linearity of simultaneous temporal and frequency discrimination using twelve subjects from a range of musical backgrounds.

First, we will discuss a little theoretical background to begin.  Signal processing is often accomplished using the Fourier Transform, which allows signals to be converted between time and frequency domains.  In terms familiar to those interested in audio reproduction, this would be the transformation between an impulse response and the frequency response, say, for a loudspeaker.  The Fourier Transformation shows that the shorter the duration is for a given function, the longer the corresponding transform will be stretched out, as well as the reverse.  This can be formalized as an uncertainty principle to show the trade off:

 ΔtΔf ≥1/4π

This linear expression sets limits on measurements of time and frequency, and as with the more widely known Heisenberg Uncertainty Principle, the expression is interpreted as meaning that the better measurement we have for one of the quantities, measurement of the other quantity will be more uncertain.  An infinitely long sound has well defined frequency content, but indeterminate time duration while conversely, an infinitesimally short burst of sound has a well defined time span, but indeterminate frequency content.  Everything else falls in between these two bounds and the limits of knowing one quantity defines the limits of how well the other quantity can be known.

To study this limit in relation to human hearing, the researchers set up a series of carefully defined tests to correlate the physical quantities of Δt and Δf with their psychological counterparts δt and δf, that define quantities of perception.  The twelve subjects were tested over a total of 38 sessions each with hundreds of test tones.  An important distinction from earlier research is the direct measurement of δt, the perception of timing, rather than conflated with measurements using physical time, Δt. 

The test regimen was based on two sets of stimulus signals, one a well defined Gaussian used to set the theoretical boundary where 4πΔtΔf = 1, and the other, an ill defined note like signal where 4πΔtΔf = 5.7079,which has a greater level of theoretical uncertainty.  The series of tests were also set up with a variable difficulty that adjusted with the test subjects performance.  The researchers used a series of five test tasks that built up in complexity, first to identify only relative frequency differences, second only identifying timing differences, and subsequent tests combining both with additional complexities.  The fifth and final task was to determine if a second tone is higher of lower in frequency than a first tone while simultaneously determining if this second tone occurs before or after a third tone that was higher in frequency than both of the other two tones.  The variable difficulty of the testing was controlled by tightening the time and frequency deltas for two correct answers in a row, but decreasing the deltas at a single wrong answer to force convergence on the test subjects limits.

Results of the testing showed that the subjects time and frequency acuity typically exceeded the Fourier Uncertainty limit by as much as a factor of 10 for the combined quantities with some displaying higher acuity of either time or frequency, but at the expense of the other quantity.  Tests based on the Gaussian tone showed most subjects exceeding 4πδtδf << 1, usually with improved timing perception at the expense of frequency perception and test subject performance with the more poorly defined note-like tone were as good of not better than with the Guassian tone.  In one case, the measured timing perception came in at a factor of 13 times the uncertainty limit at mere 3 ms that was just slightly longer than a single cycle of the test tone frequency, which had a period of 2.27 ms, but with a result of near unity for frequency perception giving a combined factor of approximately 10 times the limit.

The results of the testing regimen showed that nonmusicians typically fared as well as the musicians on the first two simple tasks where timing and frequency determination was separated, but performed more poorly on the subsequent tests that measured simultaneous perception performance.  Among the musicians, the testing showed that the composers and conductors generally outperformed players with performers only beating the uncertainly limit by a few percent on the final test while the composer and conductor group often beat the limit by as much as a factor of two.

This research has significant implications for many fields of audio processing including speech recognition and development of audio compression codecs, such as mp3, which typically assume human auditory processing functions as a bank of linear filters.  The test results also have ramifications for neuroscience in that it directly shows training as affecting and improving the limits of human perception.  Purely in audiophile terms, the results make it clear that the human ability to discern accuracy in audio reproduction cannot be adequately described using just a simple frequency response plot, as is often assumed.  Humans, particularly those with musical training, can discern much finer variations than these plots can reveal.  This is something I personally have long suspected to be the case.

While the golden-eared audiophile is sometimes a mocked concept, if he or she is a musician or has musical training, perhaps there may be something more to it than we currently understand.