(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.


The Thurston 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.
The pair 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.
After 24 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 decibel.

Fig.1 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 axis for
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 relation.

         
(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 time window.

Fig.3 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 hearing system
     The 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.
The integration 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 (1)
      (a) an impact sound only
      (b) convolution of A weight without any phase angle, and
      (c) convolution of the hearing transient response

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 Thurstonfs scale.

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 quantification I
        Abbreviations are as follows,
        H.T.R.: the hearing transient response
        M.C.C.: Multiple correlation coefficients
        I.C.: Internal correlation


Summary
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 absolutized.
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.


Remarks
(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 nerve system.
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 analysis.
Do we integrate only the part that has auto-correlation? It is an interesting question too.

(4) A few comments
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.
Evoked potential and brain wave measurements are useful information as mentioned in this chapter.

References
1) Y. Sakurai and H. Morimoto:h The transient response of human hearing systemh, J. Acoust. Soc. Jpn.(E), 10, 4(1989).
  or (2-1) of this chapter.
2) (2-1) 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 ).