Transcript Modeling Auditory Localization of Subwoofer Signals in

Applied Psychoacoustics
Lecture 4: Loudness
Jonas Braasch
Definition Loudness
Loudness is the quality of a sound that is the
primary psychological correlate of physical
intensity. Loudness is also affected by
parameters other than intensity, including:
frequency bandwidth and duration.
How can we measure loudness
• We can present a sinusoidal tone to our
subject and ask them to report on the
loudness.
Scales of Measurement
nominal
ordinal
Non-numeric scale Scale with greater
than, equal and
less than
attributes, but
indeterminate
intervals between
adjacent scale
values
e.g., color, gender
interval
ratio
equal intervals
Scale has a
between adjacent rationale zero
scale values, but point
no rationale zero
point
e.g., rank order of e.g., the
horse race finalist difference
between 1 and 2
is equal to the
difference of 101
and 102
e.g., Temperature
in Fahrenheit
e.g., the ratio of 4
to 8 is equal to
the ratio of 8 to
16.
e.g., Temperature
in Kelvin.
Measuring loudness in phon
• At 1 kHz, the loudness in phon equals the sound
pressure level in dB SPL.
• At all other frequencies the loudness the
corresponding dB SPL value is determined by
adjusting the level until the loudness is equally
high to the reference value at 1kHz (so-called
equal loudness curves or Fletcher-Munson
curves).
• Loudness measured in Phon is based on an
ordinal scale
Typical goal of our measurement
• Determine the relationship between a
physical scale (e.g., sound intensity) and
the psychophysical correlate (e.g.,
loudness)
Measurement Methods
• method of adjustment or constant response
– The subject is asked to adjust the stimulus to a
fulfill certain task (e.g., adjust the stimulus to
be twice as loud).
• method of constant stimuli
– Report on given stimulus (e.g., to what extent it
is louder or less loud then the previous
stimulus).
Equal loudness curve
Detection threshold
Loudness in Sone
• Stevens proposed in 1936 to measure
loudness on a ratio scale in a unit he
called Sone. He defined 1 Sone to be 40
phons. The rest can be derived from
measurements.
Do you have an idea how he could have
done it?
1. Measure the Sone scale at 1 kHz
e.g., by asking the subject to adjust the level of the sound stimulus
(1-kHz sinusoidal tone) such that it is twice, 4 times … as loud as
the Reference stimulus at 1 Sone (40 phons).
Relationship between Sones and Phons
At 1-kHz this is also the psychometric function between Sone and dB SPL
2. Measure the Sone scale
at all other frequencies
• The easiest way is to follow the equal
loudness contours. If they are labeled with
the correct Sone value at 1 kHz, they are
still valid.
Equal loudness curve
64 Sone
32 Sone
16 Sone
8 Sone
4 Sone
2 Sone
1 Sone
Psychometric Function
Stevens’ Power Law
Stevens’ was able to provide a general formula
to relate sensation magnitudes to stimulus
intensity:
S = aIm
• Here, the exponent m denotes to what
extent the sensation is an expansive or
compressive function of stimulus intensity.
• The purpose of the coefficient a is to adjust
for the size of the unit of measurement.
Examples for Steven’s Power Law
Examples for Steven’s Power Law
Exponents
… and now in the log-log space
An introduction to
Signal Detection Theory
Let us come back to our initial example of
determining the Absolute Threshold of
Hearing, but this time we choose a
constant stimulus approach.
How can we do it?
Measuring the ATH one more time
• For example, we can present the stimulus
at different levels ask the subject each
time whether he or she perceived an
auditory event or not.
• We might naively assume that our
psychometric function will look like this:
Number of correct responses [%]
Measuring the ATH one more time
100%
Hearing threshold
0%
Sound pressure level
… would be nice, but in reality
Probability for correct response
… it looks like this
75 % threshold
50 % threshold
Log-normalized stimulus intensity
(e.g, sound pressure level)
In signal detection theory, we explain this variation with internal noise
in the central nervous system
Definition of the correct response
Positive response
Negative response
hit
miss
Stimulus present
False alarm
Correct reject
Stimulus not present
Sometimes it is better to rather accept a false alarm (e.g., fire
detector) while other times it is better to accept a miss (e.g.,
non-emergency surgery cases)
Just noticeable differences (JNDs)
• Is the smallest value of a stimulus variation
(e.g., sound intensity) that we are able to detect.
• A classic example is the work of Fechner
measuring the JNDs for lifting different weights
Weber-Fechner’s Law
• In one of his classic experiments, Weber gradually increased the weight that a
blindfolded man was holding and asked him to respond when he first felt the
increase. Weber found that the response was proportional to a relative increase
in the weight. That is to say, if the weight is 1 kg, an increase of a few grams will
not be noticed. Rather, when the mass is increased by a certain factor, an
increase in weight is perceived. If the mass is doubled, the threshold is also
doubled. This kind of relationship can be described by a differential equation
as,
–
• where dp is the differential change in perception, dS is the differential increase
in the stimulus and S is the stimulus at the instant. A constant factor k is to be
determined experimentally.
• Integrating the above equation gives
–
• where C is the constant of integration, ln is the natural logarithm.
• To determine C, put p = 0, i.e. no perception; then
–
• where S0 is that threshold of stimulus below which it is not perceived at all.
• Therefore, our equation becomes
–
• The relationship between stimulus and perception is logarithmic
Fechner’s indirect scales
0 sensation units (0 JND of sensation)
stimulus intensity at absolute detection threshold
1 sensation unit (1 JND of sensation)
stimulus intensity that is 1 difference threshold above
absolute threshold
2 sensation units (2 JND of sensation)
stimulus intensity that is 1 difference threshold above
the 1-unit stimulus
Fechner’s Law
Determination whether difference
is perceivable
Internal response to stimulus #1,
e.g., perceived loudness
Internal response to stimulus #2
Receiver Operating Characteristic (ROC)
Discriminability index: d’=ms-mm/(s2), if ss=sm
Excitation pattern for 1-kHz sinusoid
•An excitation pattern for a 1-kHz
Sinusoid with a level of 70-dB SPL.
•The data was calculated from the
response of several single neurons.
•For each neuron, the SPL was
recorded that was needed to receive
the same discharge rate compared to
the discharge rate for the reference
condition (1-kHz, 70 dB SPL).
•The thick line plots the resulting level
of the CF tone for many neurons. Of
course, for neurons with a CF of 1 kHz,
the level has to be 70 dB, because
here the CF is identical to the reference
frequency. The further the CF is apart
from the reference frequency, the lower
the SPL that is needed to excite the
neuron with similar discharge rate,
because the neuron becomes less
sensitive to excitation at 1 kHz.
•The thin curve describes the
threshold SPL for many neurons at their
CF (Bump at 3 kHz not clear).
from Delgutte
Zwicker loudness model
Overall loudness
24
N=S
m=1
N’m
Equal Loudness Contours
dB (C)
dB (B)
dB (A), roughly 35 phons
Frequency weighting for dBA and dBC
from: Salter, Acoustics, 1998
Typical Exterior Sound Sources
from: W. J. Cavanaugh, Acoustics-General Principles, 1988
Different Weighting Schemes
from: W. J. Cavanaugh, Acoustics-General Principles, 1988
Relative Subjective Changes
Cavanaugh & Wilkens, Architectural Acoustics, 1998