(3-1) Method of acoustical estimation of an auditorium

(3-1-1) Spatial discrimination for sound field estimation
The next paper is on angle discrimination. It is taken from a few papers and figures are copied from them.
A rectangular pulse of 0.05ms was generated every one second in the anechoic chamber of our lab. A tested person was on a rotary chair and asked when he felt that coming direction was changed Seven center angles were chosen from 0 degree to 180 degrees at every 30degrees to start with. The chair was rotated clockwise and anticlockwise. The median plane was 0 degree and the loud speaker was rotated at 1.2 meters away from a tested person. It is shown in Fig.12.

Eight boy students were tested twice for each and its average was obtained. The results in Fig.22 are for the clockwise and anticlockwise. Threshold angle of discrimination for monaural hearing is given for each center angle in Fig.22 (i), and that for binaural hearing in Fig.22 (ii).
It is very interesting that the binaural hearing shows much more sensitive. The cross talk between both ears is possible to explain it.
The monaural discrimination threshold is supposed to be given by the change of head-related transfer function (HRTF). Namely, it was caused by the cross-correlation function between the HRTF at the center angle and the discriminated angle.
HRTF was measured at the eardrum of a dummy head at a different incident angle with the rectangular pulse of 0.05ms. The directivity of the HRTF at 30 degrees and 150 degrees is shown in Fig.14 as examples after it was deconvolved or normalized with the one at the front incidence. They are shown for the time domain in the left and the frequency domain in the right.

The transient response of our hearing system, usually written by R(t), was measured at the front. If R(t) is convolved with the directivity at an incident angle, the transient response of our hearing system of the angle is obtained. When directivity is expressed in the time domain being deconvolved with the front one, it can be clearer to understand its feature and it is the information enough for the direction.
As the angle discrimination was caused by the directivity of HRTF, the maximum value of the cross correlation function ψ12(τ）between the directivity at a center angle f１(t) and the one at the angle when it was felt changed, f2 (t)was discussed. It was normalized by the autocorrelation of f1(t),φ1(0)and f2(t), φ2(0)as shown in the following equation,

ψ12(τ)＝∫f1(t)f2( t＋τ)dt／[φ1(0)φ2(0)]1/2

whereφ1(0) is obtained atτ＝0 in the next equation, and the same for
φ2(0),
φ1(τ)＝∫ｆ1(ｔ)ｆ1(ｔ+τ) dt

The maximum cross correlation for each center angle is shown in Fig.15 for clockwise with ○ and counter clockwise with X. It deviates around 0.98.

The result in Fig.15 was obtained from the experiment in the horizontal level. It is evident that the discrimination is strongly depending on the directivity of the HRTF. If the directivity of a dummy head for a different angle is obtained and the space is divided with angles where cross correlation is 0.98, the spatial discrimination angle will be found.
The sound field information on the impulse response is separated in the angle and then the transient response of hearing system modified to the angle is convolved to it. The acoustic information is smoothed and easier to discuss with.

(3-1-2) Visual sound field
It is not enough to have the information only in the time domain for the evaluation of an auditorium. It must be discussed with the spatial information. Here is a try to do so which is called the visual sound field.
Reflection of a boundary accompanies edge waves and they give time and spatial information. When we discuss auditorium acoustics, a sound field must be well explained to be with. It is tried to see a sound field in the stereoscopic expression. Using the parallax, an impulse response of a sound field is observed in the space.

Some examples of boundary reflection with edge waves are given in Fig.6.
They are a large and small plane panel, a convex panel, concaved panel and a panel with reflection coefficient. Amplitude is expressed in the logarithmic scale. A positive wave is expressed with ○ and a negative with ●.

Sound fields of a rectangular room of 100x60x150cm3 are shown in Fig.3. The first reflections, the ones up to the third reflections and the ones up to the fifth reflections are given. Not only spectular reflections but also edge waves come out.

Further example is a sound field of a scale model auditorium whose reflections are up to the second. There are two pairs for two receiving points.

Here, not only many specular reflections but also a lot of edge waves are expressed.
It was shown only for an impulse response of a sound field. It must be convolved with the transient response of hearing system in the discrimination space.

An actual concert hall has complex boundaries. For the sound field calculation, it must be efficiently abbreviated. Here in Fig.5 is a preparation example of Boston Symphony Hall for the first reflection calculation.

(3-1-3) Summary for the method of acoustical estimation of an auditorium

(1) Calculation of the impulse response of an auditorium to see it in the time domain and spatially
(2) Convolution of the transient response in a discrimination angle with the directivity-modified HRTF: The space must be 0.98 on the cross correlation.
(3) Integration of its absolute value in the time window for loudness: 40ms is a temporary time window.
(4) The loudness in each discrimination angle is calculated every 40ms.
This loudness of reflections is expressed in the time sequence through the auditorium space.
(5) Using visual sound field to see the reflections in loudness, its change can be observed from one seat to another. Reputation of each seat is referred to the visual sound field.

When I got a sabbatical year in 1985 to 1986, I visited world famous concert halls for acoustical measurements. The purpose was to find good guidance for a concert hall design. Firstly the sound fields are simulated and expressed in visual sound fields, and they are compared with their reputations.
The concert halls I visited for the purpose were as follows:

Bynjanei Haoma(Israel)、
Stadt Casino (Switzerland)、
Grosser Music Vereinssaal(Austria)、
Concert Gebau(Holland)、
Boston symphony Hall(USA)、
Tanglewood Music Shed (USA),
Caracas(Venezuela)、
Teatro Colon (Argentine)