Human Time-Frequency Acuity Beats the Fourier Uncertainty Principle

The time-frequency uncertainty principle states that the product of the temporal and frequency extents of a signal cannot be smaller than $1/(4\pi )$. 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.

Figure 1

Stimulus and task. (a) In our final task 5, subjects are asked to discriminate simultaneously whether the test note (red) is higher or lower in frequency than the leading note (green), and whether the test note appears before or after the flanking high note (blue). For each instance of the task, two numbers are generated (Dt and Df) and two Boolean responses (left-right, up-down) are recorded. (b) Tasks 1 through 4 lead to this final task: task 1 is frequency only (uses two flanking notes), task 2 timing only, task 3 is frequency only but with the flanking high note (blue) as a distractor, and task 4 is timing only, with the leading (green) note as a distractor [24].

Figure 2

(a) Measurement strategy. As task 5 proceeds, the numbers Df and Dt are drawn as Gaussian random numbers with variances pref and multf. The smaller these variance, the harder the task. The variances are independently controlled by a 2D1U (two down, one up) procedure: when two responses in a row are correct, the variance is reduced and the task is made harder; the variance is increased for every wrong response. This procedure converges to a demanding regime, where the subject makes frequent mistakes, but fewer than 50%. Data shown are from subject $\mathbf{q}\mathbf{r}\mathbf{3}\mathbf{z}\mathbf{b}$ [24]. (b) Datum definition. We show in red the time responses of subject $\mathbf{q}\mathbf{r}\mathbf{3}\mathbf{z}\mathbf{b}$; horizontal axis is Dt, vertical axis is 0 (for before) or 1 (for after); we have slightly offset the data by random amounts from 0 or 1 to better visualize the density of points at any given Dt. In blue, the psychometric curve which maximizes the likelihood of the data. The procedure described in 1(b) has converged to a high density of tests around 0, spanning the steepest area of the psychometric curve.

Figure 3

Summary of main results: discrimination limens for each test. Each round dot is a completion of Task 5 by a subject on an individual day, with at least 100 presentations. There were 12 subjects totaling 26 individual sessions for Gaussian and 12 sessions for notelike tests. Blue denotes a Gaussian packet while red denotes a notelike packet. The two solid lines are the locus of the relation $\delta t\delta f=\Delta t\Delta f$; any dots below these curves violate the corresponding uncertainty relation. Error bars in both dimensions were obtained by generating 1000 bootstraps from the raw data and plotting the 25–75% quartiles. Raw data provided in the Supplemental Material, Table S1 [24], see also Ref. 28.