(2-2) Loudness of an impact sound
with a time window
The linear part of our hearing system was
found1) to have a pair comparison of the loudness of two rectangular
pulses with that of a single rectangular pulse, changing their time intervals.
The rectangular pulse had 0.05ms time duration, which covers our audio
frequency, and had the amplitude of 93dB or 87dB.
This linear response is supposed
to include the head related transfer function (HRTF), the elastic response of
the eardrum to the three little bones, the lymph liquid in the cochlea, and the
elastic movement of the basilar membrane. It might include even a part of
peripheral nerve system.
It was shown that there is a process
to make a sound absolutized2) after the linear process.
It is discussed in this section how the
loudness of an impact sound is decided at the higher level. It is tried to find
how it weighs on frequency and forms a time window.
scale by pair comparison for various impact noises
First, eight kinds of impact noises were
recorded with slight adjustment as shown in Fig.1. Each impact noise got three
different levels with an 8dB step and 24 impact noises were prepared. Three
peak levels for an impact noise are given in Fig.1.
comparison was done in the echoic chamber by 18 tested persons. At pair
comparison, another pair was given at 3.5 seconds after the first impact noise
was ceased1). The combination for a pair was not reversely done and
it was the pairs of 24x23/2.
impact noises got the pair comparisons, they got the Thurstonfs scale in the
case V and were arranged on an axis as in Fig.2.
The linear response of our
hearing system and its absolutization process are shown previously. The next
process is supposed to have the integration in a time window. It is expressed
as in the equation3),
where P (t) is a given sound pressure which
is an impact noise in Fig.1. R (t) is a transient response of hearing system to
be convolved. shows convolution
product. t1 to t2 is the interval of a time window for
the integrand. Fo pis a function of power or
logarithmic. Here the latter is used practically 20log10 to get a
Twenty-four impact sounds used for the pair comparison
Fig.2 Loudness on
the Thurstonfs scale for the 24 impact sounds in Fig.1
For the function R (t) in Eq. (1)
to convolve to an impact noise, (a) 1 to have the physical pressure itself, (b)
A - weight without any phase angle and (c) the transient response of our
hearing system1) were given. After then they were absolutised and
integrated for 40ms as an example.
A decibel value is obtained for
an integrated value. The decibel value of the Eq. (1) is given on the
horizontal axis and the corresponding Thurstonfs scale is given on the vertical
each R(t) as shown Fig.3.
Three different levels of each impact noise
are connected with a line. The decibel value of the integration of Eq. (1) correlates
to the Thurstonf scale at each R (t). It shows that the logarithmic scale has
been well used for estimating the loudness level of a sound.
In each figure, each line is shifted almost
in parallel. It suggests that there exist independent factors to explain the
(a) Integration of an impact
sound itself in the 40ms time window.
(b) Integration of an
impact sound convolved with A - weight in the 40ms time window.
(c) Integration of an impact
sound convolved with the transient response of our hearing system in the 40ms
Thurstonfs scale vs. integration of Eq. 1 after the convolution (a) of an
impact sound only, (b) with A - weight and (c) with the transient response of
vertical axis shows the Thurstonfs scale and the horizontal the decibel value
calculated by Eq. (1). Ⓐ`Ⓗ shows an impact sound in Fig. 1.
Each has three different levels.
To find a time window with the application of the
theory of quantification I
Having the Thurstonfs scale as an outsider
and a variety of time window as a factor, the theory of quantification I was
applied to see how a multiple correlation coefficient changes.
It was searched which time window shows the
largest multiple-correlation coefficient. It must be the most proper time window.
An impact noise is convolved as it was done
in the previous section: (a) just one to have a physical impact noise, (b) A
weight without any phase angle and (c) the transient response of our hearing
system. After each convolution, it is absolutized and integrated with a variety
of time window.
with a time window converted to the decibel value with 20log10. It was categorized with a 5dB step. Categories for
convolutions (a) (b) and (c) in the above are varied 7 to 9. The
result is show in Table 1.
Table 1 Multiple correlation coefficients by
the theory of quantification I for three different convolutions in the integral
(a) an impact sound only
(b) convolution of A weight without
any phase angle, and
(c) convolution of the hearing transient
It shows that the largest multi-correlation
coefficient among convolving functions for every time window was with the
transient response of hearing response. Namely, it says that it is most proper
for weighing to our hearing system.
And at the time window of 40ms the
multiple-correlation coefficient is the largest for the outsider of the
Selected impact noises in the left column of
Fig.1 include pure tones. It was quantified with the theory of quantification
I, having another factor for pure tone. Multiple correlation coefficients are
shown in Table 2 with the internal correlation coefficients.
Unfortunately they were relatively large and
the results were not reliable to see with time window.
At the time window 40ms the internal
correlation coefficient is small with0.087. But the multiple-correlation
coefficient changed only 0.9322 to 0.9357. It is not affected by the pure tone
factor. It might have been accepted as a part of the noise.
Possible additional factors should be
discussed considering the parallel shift in Fig.3 (c) and the multiple-
correlation coefficient 0.9322.
Table 2 Multiple correlation coefficients for
two factors of time window and pure tone component at the theory of
Abbreviations are as
H.T.R.: the hearing transient
The loudness of an impact noise is estimated as follows:
Firstly, the transient response of our hearing
system is convolved to a given impact noise. If it has an incident angle, its
directivity is convolved too.
Before each signal comes to binaural hearing it is
Meantime, a path way is chosen for the signal;
For instance, a pure tone from
outside does not beat with a low pitch sound, the resonance frequencies are
smoothed at the transient response for the rectangular pulse of 0.05ms. Path
ways for a rectangular pulse and a pure tone seems to be different.
A time window is supposed to be decided by the
information and the auto-correlation of a given signal, and the integrand of Eq.
(1) is integrated during the time window.
40ms is the best time window for an impact noise.
The non-linear function of power or logarithmic are
supposed to be made with the saturation of excessive large input and the
internal noise of our self, and the loudness level is given.
This result is given in Eq. (1) which
is rewritten below and the time window t1 to t2 is 40ms
for an impact noise.
(1) Absolutization in the hearing system and sound
field estimation with intensity or energy density
The estimation of a sound is done with the
loudness that is obtained from Eq. (1) with the time window of 40ms. This attitude
of listening sound is done to every direction convolved with the directivity as
mentioned before. It is absolutized there.
If we have double amplitude of
sound from the front, we hear it doubled referring to the loudness of the
single. If sound comes from a different angle, the loudness is depending on the
directivity of the HRTF. If we get } rectangular pulses, we hear as if we got two positive pulses.
This attitude to listen sound can
understand how energetic treatment in a sound field explains well a noise
environment. However, it is meaningful to research the points where we canft
wholly do it. A diffusive sound field does not have any particular direction to hear.
(2) Time window for a 0.05ms rectangular pulse
We learnt that two
rectangular pulses of 0.05ms are heard as the same loudness at the time
interval of 3.5-3.8ms. It means that the time window
for non-correlated two pulses is finished to have integration. This must be the
shortest time window. It starts to be separated at 1.4 to 1.7ms but it is not yet done by
the time window.
Even after this shortest time
window, it is not enough long to understand the meaning of the signals. They
are not auto-correlated for that.
(3) Time window after 40ms
The time window of 40ms is for the
peripheral processing of a relatively short sound. The central processing, like
understanding of language or music listening, starts after then.
The path difference with 40ms is
13.6m. For 50ms itfs 17m. It is often referred this path difference to have an
echo especially on speech.
A time window would be different
if a sound is used to hear or not.
The auto-correlation of a sound must be
found and its loudness is found how much it is related to it, using the above
mentioned methodology with the Thurstonfs scale.
These discussions are before the peripheral
A time window would be different
depending on the content or the information of a sound because one tries to
catch the meaning of the sound having auto-correlation
When we listen to words or music,
which has a meaning, a time window depends on experience, its information, time
change, kinds, all will be related to it. After all, we need the multivariable
Do we integrate only the part that has auto-correlation?
It is an interesting question too.
(4) A few
It is very fundamental to see the loudness
change when the rectangular pulse width is changed. It is convolved with the
transient response and does not show monotonous loudness increase. Once it
must be done.
Loudness or annoyance estimation, listening to a
talk, listening music, each attitude is different. The discussion must be done
separately for each.
It must be interesting to discuss with other
sensory systems referring with each other. It is mentioned a bit in Chap. V. There discussions must be separately done
for short time one, e.g. sensation, and long time one, e.g. memory or
experience, because the upper processing is supposed to be different.
potential and brain wave measurements are useful information as mentioned in
1) Y. Sakurai and H. Morimoto:h The
transient response of human hearing systemh, J. Acoust. Soc. Jpn.(E), 10,
or (2-1) of this chapter.
of this chapter.
3) Y. Sakurai et al:hLoudness of a given impact soundh,
p209-212, Kinki Branch Meeting of the A.I.J. (1990) ( in Japanese language ).