Amir and prof if you read this article what does it means for the possibility you claimed erroneously as a FACT : the reduction of the non linear non symmetrical time domain ears/brain way to extract information to simple electrical linear modeling tool in the symmetrical physical time domain ?
Do you think it is possible ?
No it is not.... Human ears/brain non linearities structure and internal "tools" beat the Gabor limit if not in resolution in precision in the time domain......
Then why claiming that your tools can decide what is perceived and what is not "a priori" by someone listening chorus music in an acoustically controlled room ?
How can you claim A PRIORI, with your simple measuring specs tools designed for gear standars verification, that no change will be perceived at all by changing some materials parameter, some gear component, or some acoustical modifications ?
How can you claim this using electrical tool working in the electrical linear modeling symmetrical domain, if the ears is able to extract "precise tracking" information and change in the non symmetrical time domain ?
Here is the original non vulgarized article beginning :
«Human Time-Frequency Acuity Beats the Fourier Uncertainty Principle
Jacob N. Oppenheim and Marcelo O. Magnasco∗
Laboratory of Mathematical Physics, Rockefeller University, New York, New York 10065
(Dated: March 13, 2015)
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.
PACS numbers: 43.60.+d,43.66.+y,87.19.L-
Fourier transformation turns signals “inside out”, in
the sense that low frequencies dictate what happens at
long times, while high frequencies create fine temporal
detail. This property is demonstrated by Fourier’s un-
certainty theorem, which states that considering the ab-
solute value squared of a signal x(t) as a probability dis-
tribution in time,
P (t) = |x(t)|2
∫ ∞
−∞ |x(t′)|2dt′ (1)
and the absolute value squared of its Fourier transform
˜x(f ) as a distribution in frequency,
P (f ) = |˜x(f )|2
∫ ∞
−∞ |˜x(f ′)|2df ′ (2)
then the product of the standard deviations
∆t = √var(t) and ∆f = √var(f ) (3)
is bounded from below [1]:
∆t∆f ≥ 1
4π (4)
whence it is inferred that short signals require many fre-
quencies for their representation.
The theorem refers to the original signal and its Fourier
transform. In time-frequency analysis one attempts to
describe a signal in the two-dimensional time-frequency
plane, akin to a musical score where time is the horizontal
axis and frequency the vertical axis. Here the uncertainty
principle begets the Gabor limit [1, 2]. This remapping
emphasizes the uncertainties as a property of the trans-
form itself, rather than the the signal. In time-frequency
analysis, it has been proven that linear operators can-
not exceed the uncertainty bound [2]. Nonlinearity does
not by itself confer any acuity advantage, and in fact
most nonlinearities are merely distortions and thus dele-
terious. However, by the above theorem, any carefully-
crafted analysis that can beat this limit must necessarily
be nonlinear. For instance, precise frequency informa-
tion can be obtained about a sine wave by measuring
the time between two adjacent zeros of the waveform,
a clearly nonlinear operation. The nonlinear distribu-
tions can be classified in families according to their de-
gree of nonlinearity or history-dependence, such as the
quadratic (Cohen’s class) distributions like Wigner-Ville
[3] and Choi-Williams [4], and higher-order ones, such
as multi-tapered spectral derivatives [5, 6], the Hilbert-
Huang distribution [7], and the reassigned spectrograms
[8–12]. To understand how they differ we need to make
an important distinction between resolution and preci-
sion. Resolution refers to our ability to distinguis two
objects, while precision refers to our ability to track the
parameters of a single object, given prior knowledge it is
only one component. This distinction is well-established
in optics, where it is known the wavelength of light limits
resolution: two glass beads cannot be resolved as different
in a microscope if they are closer together than a wave-
length. Precision is not limited, since a single bead can be
tracked with nanometer accuracy. All the above distribu-
tions achieve higher precision than the Gabor limit when
applied to isolated signal components, yet give interfer-
ing results when two signals are closer together than an
uncertainty envelope. Our experimental test is designed
to directly measure precision, not resolution.
A key goal in neuroscience is to establish which algo-
rithms the brain uses to process perceptual information.
Psychophysics, by establishing tight bounds on the per-
formance of our senses,may rule out entire families of
perceptual algorithms as candidates when they cannot
achieve the expected performance [13, 14].
We shall show below that human subjects can discrim-
inate better, and occasionally much better, than the un-
certainty bounds. This categorically rules out any first
order operators, such as the standard sonogram, from
consideration, and puts a stringent bound on the perfor-
mance of any candidate algorithm, demonstrating that
the nonlinearities in the cochlea constitute are integral
to the precision of auditory processing.»..........
«A key goal in neuroscience is to establish which algo-
rithms the brain uses to process perceptual information.
Psychophysics, by establishing tight bounds on the per-
formance of our senses,may rule out entire families of
perceptual algorithms as candidates when they cannot
achieve the expected performance [13, 14].
We shall show below that human subjects can discrim-
inate better, and occasionally much better, than the un-
certainty bounds. This categorically rules out any first
order operators, such as the standard sonogram, from
consideration, and puts a stringent bound on the perfor-
mance of any candidate algorithm, demonstrating that
the nonlinearities in the cochlea constitute are integral
to the precision of auditory processing.
The conclusion of this article :
«We have conducted the first direct psychoacoustical
test of the Fourier uncertainty principle in human hear-
ing, by measuring simultaneous temporal and Our data indicate that human subjects
often beat the bound prescribed by the uncertainty the-
orem, by factors in excess of 10. This is sometimes ac-
complished by an increase in frequency acuity, but by and
large it is temporal acuity that is increased and largely
responsible for these gains. Our data further indicate
subject acuity is just as good for a note-like amplitude
envelope as for the Gaussian, even though theoretically
the uncertainty product is increased for such waveforms.
Our study directly rules out many of the simpler models
of early auditory processing, often used as input to the
higher-order stages in models of higher auditory function.
Of the plethora of time-frequency distributions and au-
ditory processing models that have been studied, only a
few stand a chance of both matching the perfrequency
discrimination. formance of
human subjects and be plausibly implementable in the
neural hardware of the auditory system(e.g.[6, 7, 12, 28],
with the reassignment method having the best compara-
tive temporal acuity. Elucidation of which mechanism
underlies our subjects auditory hyper acuity is likely
to have wide-ranging applications, both in fields where
matching human performance is an issue, such as speech
recognition, as well as those more removed, such as radar,
sonar and radio astronomy.»
https://arxiv.org/pdf/1208.4611.pdf
