ISRA2013

Toronto, Canada International Symposium on Room Acoustics

2013 June 9-11

1

Analysis method for estimating diffuseness of sound fields by using decay-cancelled impulse response

Toshiki Hanyu (hanyu@arch.jcn.nihon-u.ac.jp) Department of Architecture and Living design Nihon University, Junior College 7-24-1 Narashinodai, Funabashi, Chiba 274-8501, Japan

ABSTRACT

An analysis method for estimating diffuseness of sound fields by measuring the time variation in reflected sound energy of impulse responses is proposed. In this method, first a decay-cancelled impulse response is obtained by removing the reverberation decay from the impulse response using a Schroeder decay curve. The degree of diffusion of the sound field is determined by evaluating the time variation in the reflected sound energy of the decay-cancelled impulse response. By using this method, the frequency characteristics of diffuseness in sound fields can be analysed from the impulse response measured at a single point. The average degree of diffusion in a room can also be evaluated by averaging the analysis results at several points in the room, similar to the analysis of reverberation time.

In order to verify the proposed method, the impulse responses in rooms having different types of diffusers were calculated by the wave acoustics computer simulation. The frequency characteristics of diffuseness were analysed from the calculated impulse responses, by using the proposed method. The results showed that the frequency characteristics of diffuseness change depending on the size of the diffusers. Thus, the proposed method can be used for evaluating the effect of diffusers on the degree of diffusion in a sound field.

1 INTRODUCTION

Regarding room acoustic design, the scattering capabilities of wall surface is as important a design factor as absorption of walls, particularly with regards to the ensuing diffusion of the sound field inside the space. Therefore, measures of evaluating the wall surface diffusion, such as a scattering coefficient and a diffusion coefficient, have been defined[1, 2]. And the author has presented a theoretical framework for quantitatively characterizing sound field diffusion based on scattering coefficient of walls[3]. On the other hand, a measurement method for estimating diffuseness of sound fields has not been systematized. The measurement method of sound field diffuseness is necessary in order to relate the wall surface diffusion to diffusion of the sound field in actual sound fields.

Jeon et al. proposed a measurement method which counts local peaks that exceed -20dB relative to the direct sound level in an impulse response for evaluation of the diffusion of sound fields[4]. This is a simple method to evaluate the effect of wall surface diffusion because the number of reflected sound increases if diffusers are attached to the wall surfaces. And easiness of measurement is one of advantages of this method. However the result of counting peaks

2

depends on the direct sound level, i.e. a receiver position, reverberation time of the sound field, and the measurement system like used loud speaker. Therefore it is difficult to compare results of measured diffuseness between different sound fields. And this method cannot analyse the frequency characteristics of diffuseness.

The main purpose of this study is to develop an analysis method for estimating frequency characteristics of diffuseness of sound fields. This paper focuses especially on how to detect the time fluctuation in reflected sound energy of impulse responses.

2 OUTLINE OF PROPOSED METHOD

2.1 Decay-cancelled Impulse Response

Schroeder decay curve is calculated as follows:

tS dptE )()( 2 . (1)

By using this Schroeder decay curve, a decay-cancelled impulse response is defined by the following equation.

)(

)()(

tE

tptg

S

(2)

And a squared decay-cancelled impulse response is expressed as follows:

tS dp

tp

tE

tptg

)(

)(

)(

)()(

2

222 . (3)

The numerator is a squared impulse response and the denominator is a Schroeder decay curve. Here in the meanwhile, if it is assumed that the detail of time structure of the impulse response is ignored and that the energy decay is accordance with the exponential law with decay rate A, squared impulse response can be modeled as follows:

AtPtp exp)( 22 (4)

where P represents maximum amplitude of the impulse response. In this condition, the Schroeder decay curve can be expressed by the following equation.

AtA

P

dAPtEtS

exp

exp)(

2

2 (5)

3

Substituting Equations (4) and (5) to Eq.(3), the following relationship can be obtained.

A

AtA

P

AtPtg

exp

exp)(

2

22 (6)

When time fluctuation of the impulse response is ignored, g2(t) becomes the same as the decay rate A. Because there is an energy fluctuation of the impulse response in an actual sound field, an average value of g2(t), )(2 tg , becomes the same as A.

Atg )(2 (7)

As described above, a decay-cancelled impulse response can be calculated by using a Schroeder decay curve, and sound energy decay rate of a sound field can be estimated by using an average value of a squared decay-cancelled impulse response, )(2 tg .

t

Es(t)

t

p2(t)

tA

g2(t)P2

P2

A

Figure 1: Conceptual diagram of squared decay-cancelled impulse response, Schroeder decay curve and squared decay-cancelled impulse response

2.2 Method for Estimation of Reverberation Time using a Decay-Cancelled Impulse Response

When T represents reverberation time, the following relationship can be obtained.

AT explog1060 10 (8)

By using this relationship reverberation time can be obtained by using decay rate A as follows:

A

AeT

82.13

1

log

6

10

. (9)

4

Substituting Eq.(7) into Eq.(9), the following equation can be obtained.

)(

82.132 tg

T (10)

Eq.(10) means that reverberation time can be analyzed from the average value of a squared decay-cancelled impulse response, )(2 tg .

2.3 Normalized Decay-cancelled Impulse Response

In nature diffuseness of sound field should be evaluated by both uniformity of spatial distribution of reflected sound energy and uniformity of propagation direction. Both uniformities are thought to be reflected to an energy fluctuation of each impulse response in the sound field. In a sound field with low degree of diffusion, time fluctuation of reflected sound energy in each impulse response must be large. Based on this assumption, the proposed method indirectly evaluates degree of diffusion by evaluating time fluctuation of reflected sound energy in the impulse response.

g2(t) can be interpreted as one which detects only time fluctuation of reflected sound energy by cancelling the energy decay. However because a magnitude of g2(t) is different when a reverberation time is different, the fluctuation cannot be compared by using g2(t) as is. Therefore a normalized decay-cancelled impulse response h(t) is defined as the followings.

)()(

1)(

2tg

tgth (11)

By this normalization, )(2 th becomes always 1 and the time fluctuation of reflected sound energy of an impulse response can be compared between different sound fields.

1)(2 th (12)

t

g2(t)

t

)(2 th

Ttg

82.13)(2 1)(2 th

Figure 2: Squared decay-cancelled impulse response g2(t), square of normalized decay-cancelled impulse response h2(t), and their average

5

2.4 Fluctuation Decay Curve of Reflected Sound Energy

A squared inpulse response p2(t) means a response in dimension of energy. On the other hand h2(t) means the time fluctuation of relative magnitude of sound energy to the average energy decay curve of a target sound field. If the time fluctuation of reflected sound energy in an impulse response is small, h2(t) fluctuate near 1.

Therefore based on this, an analysis method for estimating diffuseness of sound fields is investigated. In evaluation time range, t1～t2, the total of h2(t) is defined as Rtotal.

dtthRt

ttotal 2

1

2 (13)

On the other hand a total of h2(t) when the value exceeds threshold, k, in the evaluation time range is defined as R(k). And the ratio z(k) is defined as the followings:

totalR

kRkz

(14)

This ratio z(k) means probability that relative magnitudes of sound energy of the impulse response to the average energy decay curve exceed the threshold k. As shown in Figure 3, the z(k) decreases as the threshold k increases, and z(k) becomes a kind of decay curves. Therefore this z(k) is named as the “fluctuation decay curve”. A steeper fluctuation decay curve means that the time fluctuation of the sound energy in the impulse response is small. Therefore diffuseness of sound field might be evaluated by the fluctuation decay curve.

k kR

totalR

減衰除去二乗応答h2(t)

t1 t2

)(2 th

Evaluation time range

t

Figure 3: Method for obtaining fluctuation decay curve of reflected sound energy z(k) from h2(t) by using threshold k

2.5 Degree of Time Series Fluctuation of Reflected Sound Energy

As shown in Figure 4, the threshold k at which fluctuation decay curve z(k) becomes 0.01 is defined as the "degree of time series fluctuation" of reflected sound energy. A steeper fluctuation decay curve indicates smaller degree of time series fluctuation. The degree of time series fluctuation indicates how large the reflected sound energy where probability of

6

occurrence is 1%. Therefore, smaller degree of time series fluctuation means higher degree of diffusion of sound field.

0 20 40 60 80 100

z

Threshold of Energy Ratio（ｋ）

13 21 72

z(k)

threshold k

10-2

Figure 4: Examples of fluctuation decay curves of three different sound fields and degrees of time series fluctuation of them

3 VERIFICATION BY WAVE ACOUSTICS COMPUTER SIMULATION

3.1 Calculation Conditions

Because two-dimensional FDTD was used as wave acoustics computer simulation method, verification has been done using two-dimensional space. As shown in Figure 5 calculation target room was a circular room whose diameter is 25.4m. Conditions of diffusers installed on circular wall were three patterns as follows:1) without diffusers, 2) with small diffusers, and 3) with large diffusers. So a total of three conditions were used. Absorption coefficient of the wall in all patterns was 0.1. Impulse responses at six receiving points shown in Figure 5 were calculated. Calculations have been done under the following conditions: 1) Sampling frequency: 50 kHz, 2) intervals of calculation mesh: 1cm. After the FDTD calculation, obtained impulse responses were filtered in order to obtain responses in octave bands from 63 Hz to 2 kHz. And the degree of time series fluctuation was analysed from each octave band response.

25.4m

Circular room

Sound source

Receivingpoints

Width : 0.4mHeight : 0.06m

Small diffuser

Width : 1.5mHeight ： 0.25m

Large diffuser

Figure 5: Target room for the computer simulation

7

3.2 Calculation Results

Figure 6 shows impulse responses calculated in three conditions of wall surface treatment. As shown in this figure, wave forms of impulse responses change depending on the wall surface treatment. Especially the wave forms of impulse responses in the rooms with small or large diffusers are smoothed comparing to the impulse response in the room without diffusers.

Figures 7 and 8 show the analysis results of degree of time series fluctuation in each octave band. Figure 7 shows the comparison between the following two conditions: 1) without diffusers and 2) with small diffusers. Figure 8 shows the comparison between the following: 1) without diffusers and 3) with large diffusers. All of these results were calculated by averaging the results of six receiving points.

When looking at the result of "without diffusers" in Figure 7, the degree of time series fluctuation is high at higher frequency band. This result indicates that degree of diffuseness of this sound field is low at higher frequency. Comparing the result of "with small diffusers" to the result of "without diffusers" in Figure 7, it can be seen that the degrees of time series fluctuation above 500Hz bands decrease due to the small diffusers. Furthermore, comparing the result of "with large diffusers" to the result of "without diffusers" in Figure 8, the degrees of time series fluctuation above 125Hz bands decrease due to the large diffusers. These results indicate that the frequency where diffusers affect sound field diffusion becomes the same as the frequency where the wave length of the sound is equal or short to diffuser's width. These results are coincident with a well-known phenomenon of sound scattering.

1) without diffusers

2) with small diffusers

3) with large diffusers

Figure 6: Impulse responses calculated in three conditions of wall surface treatment

8

1

10

100

1000

63 125 250 500 1k 2k

時系列ば

らつき度（非拡

散度）

オクターブ中心周波数[Hz]

拡散体なし

拡散体小

Deg

ree

of ti

me

seri

es f

luct

uati

on

Octave band center frequency [Hz]

Frequency where wave length of the sound is equal to diffuser’s width

with small diffusers

without diffusers

Figure 7: Comparison between analysis results of degree of time series fluctuation in conditions: 1) without diffusers and 2) with small diffusers

Frequency where wave length of the sound is equal to diffuser’s width

With large diffusers

without diffusers

Deg

ree

of ti

me

seri

es f

luct

uati

on

Octave band center frequency [Hz]

Figure 8: Comparison between analysis results of degree of time series fluctuation in conditions: 1) without diffusers and 3) with large diffusers

9

4 CONCLUSIONS

An analysis method for estimating diffuseness of sound fields by measuring the time fluctuation in reflected sound energy of impulse responses is proposed.

1) First a decay-cancelled impulse response is obtained using a Schroeder decay curve.

2) The degree of diffusion of the sound field is determined by evaluating the time fluctuation in the reflected sound energy of the decay-cancelled impulse response.

3) By using this method, the frequency characteristics of diffuseness in sound fields can be analysed. The average degree of diffusion in a room can also be evaluated by averaging the analysis results at several points in the room, similar to the analysis of reverberation time.

In order to verify the proposed method, the impulse responses in rooms having different types of diffusers were calculated by the wave acoustics computer simulation. The results showed that the frequency characteristics of diffuseness change depending on the size of the diffusers. Thus, the proposed method can be used for evaluating the effect of diffusers on the degree of diffusion in a sound field.

REFERENCES

1 ISO 17497-1:2004: Acoustics - Measurement of the sound scattering properties of surfaces, Part 1: Measurement of the random-incidence scattering coefficient in a reverberation room (2004).

2 AES-4id-2001, “AES Information document for room acoustics and sound reinforcement systems – characterization and measurement of surface scattering uniformity,” J.Audio Eng.Soc., 49, 149-165, (2001).

3 T. Hanyu, “A theoretical framework for quantitatively characterizing sound field diffusion based on scattering coefficient and absorption coefficient of walls,“ J. Acoust. Soc. Am., 128, No. 3, 1140–1148 (2010.9)

4 Jin Yong Jeon, Yong Hee Kim, and Michael Vorländer, “Counting local peaks in impulse responses for evaluation of the in‐situ diffusion in concert halls.,” J. Acoust. Soc. Am. Volume 129, Issue 4, pp. 2501-2501 (2011)