Comment on "Human time-frequency acuity beats the Fourier uncertainty principle".

it was claimed that human hearing can beat the Fourier uncertainty principle. In this Comment, we demonstrate that the experiment designed and implemented in the original article was ill-chosen to test Fourier uncertainty in human hearing. The Gabor limit [1], ∆t∆f ≥ 1 4π , (1) refers to the lower bound on the product of the standard deviations (STD) in time (∆t) and frequency (∆f) of an audio signal. This limit is a consequence of the Fourier uncertainty principle. In their Letter, Oppenheim and Magnasco [2] claim that human hearing can surpass this limit. They design an experiment which establishes psychological limens, δt and δf , and show that their subjects can discriminate signals that beat a limen-based uncertainty δtδf ≥ 1 4π. (2) The frequency and time limens used by the authors relate to the accuracy with which human participants can distinguish small frequency and time shifts present in a sequence of three test pulses. The sequence in question is referred to as " Task 5 " in the paper. It is our view that their experiment is ill-chosen to test Fourier uncertainty. Firstly, the ∆t and ∆f that appear in the Gabor limit must be the STD of time and frequency evaluated over the whole test signal. This point is made clear in the derivation of the uncertainty principle that can be found in the book of Cohen [3] (Sections 3.2 and 3.3). The li-mens used by the authors, however, are simply ad hoc parameters that relate to the STD of statistical errors made by the human participants when tasked with estimating frequency and timing shifts in the test signal, and are unrelated to the STD of time and frequency evaluated over the test signal. Therefore, the limen-based inequality Eq.(2) is in no way related to the actual Gabor limit Eq.(1), and there is no expectation that the limen-based inequality should be satisfied. Secondly, one can straightforwardly use Fourier analysis itself to " beat " Task 5, which again demonstrates that Task 5 does not test for violations of Fourier uncertainty since any Fourier-based analysis would necessarily be limited by the uncertainty principle. One method is to use a window Fourier Transform (WFT) to construct a spectrogram given by