This article is an article submitted by the author to the "21dB Acoustics" public account, and is now permitted to be reproduced on my blog. For technical exchanges, please contact [email protected]
In a closed space, when the sound travels from the sound source to the receiving point, there is not only the direct sound, but also the reflected sound reflected by the wall multiple times. Therefore, after the sound source stops sounding, the sound in the room can still be heard by the human ear for a period of time. This phenomenon is defined as reverberation. The time required for the sound pressure level to drop by 60 dB after the sound source stops sounding is defined as the reverberation time [1] (Reverberation Time, T 60 T_{60}T60). Reverberation time is an important parameter to characterize the acoustic characteristics of a room, which is mainly determined by the size of the room and the acoustic characteristics of the surface material.
Room reverberation can cause spectral coloring and smearing of sound signals, thereby affecting human perception of sound signals. During this process, the fine structure of the sound signal and its envelope are changed. The optimum reverberation time depends on the purpose of the room [2] (e.g. conference, music, theater, etc.). For an ordinary room, if the reverberation time is too long (greater than 2 s), the room is called "echoic", and the sound appears wet and muddy. If the reverberation time is too short (less than 0.3 s), the room is said to be "acoustically dead" and the sound appears dry and unnatural. The design of room reverberation uses the principles of room acoustics and psychoacoustics, and requires sound acoustic theory and a lot of practical experience to support it. Generally speaking, the best reverberation time in recording studios and broadcasting studios is required to be below 0.5 s [3]. This is to reduce the impact of the room on the sound, which is more conducive to the later sound processing. The optimal reverberation time of the classroom is required to be below 1 s [4]. For example, the classroom reverberation time specified in the classroom acoustics standard in the United States is 0.6-0.8 s, because speech intelligibility and speech quality depend on the fine structure and envelope of speech. When the reverberation is too long, phonemes are blurred or covered, and speech intelligibility will be seriously affected [5]. Being in an environment with too long reverberation time for a long time will cause students' hearing fatigue, which will lead to a decline in the quality of teaching. At the same time, for hearing-impaired listeners, excessive reverberation will also aggravate their hearing problems in a complex acoustic environment. The optimal reverberation time for lecture halls and cinemas is around 1 s. The best reverberation time of large theaters and concert halls is often above 1.5 s, because longer reverberation time will make the music round and dynamic, full of sense of space [6].
Figure 1 is a typical room system. In a closed room, the room sound field can be solved analytically by solving the wave equation. However, this method is based on an ideal model assumption and has high computational complexity, making it difficult to be directly applied to real rooms with complex shapes. In actual acoustic design, the details of individual modes and rays are often not concerned, but the energy distribution of the sound field of the entire room. Therefore, the room reverberation time calculation method based on statistical acoustics is mainly used. Statistical acoustics assumes that sound waves have the same probability of propagating in all directions, contribute the same energy to the sound field, and have an irregular phase. The sound field in which the energy density is uniformly distributed in space is called diffuse sound field [1], which is an important assumption in statistical acoustics. At the end of the 19th century, Sabine established the empirical formula of the room reverberation time under the diffuse sound field through acoustic experiments, that is, the Sabine formula:
RT 60 = 0.1611 V ∑ α k S k R T_{60}=0.1611 \frac{V}{\sum \alpha_{k} S_{k}}RT60=0.1611∑akSkV
Among them, VVV is the volume of the room (in m3),S k S_{k}Skis the surface area of the first sound-absorbing material in the room (in m2), α k \alpha_{k}akis the absorption coefficient of the first sound-absorbing material in the room. As can be seen from the formula, the room reverberation time depends on the room size and the sound absorption coefficient of the material. In the same room, if the sound absorption coefficient of the material is increased, the reverberation time of the room will decrease and the sound will die out more quickly. The Sabine formula was first obtained by many acoustic experiments, and later verified by statistical acoustic theory. To this day, Sabine's formula is still the standard formula for calculating reverberation time. The sound absorption of the surface of the room generally depends on the acoustic properties of the material. The acoustic material absorbs different sound waves at different frequencies. Therefore, the reverberation time of the room often needs to be calculated by sub-band. The Sabine formula does not consider the shape of the room and the loss of sound when it propagates in the air, and the absorption of high-frequency sound waves by the air in a large room cannot be ignored, so the Sabine formula is consistent with the actual high-frequency reverberation time calculation. The situation is different.
Subsequent research work revised the Sabine formula [7]. For the calculation of the reverberation time of a small room with a large sound absorption coefficient, Eyring proposed the famous Eyring formula:
T = 0.161 V − S ln ( 1 − α ˉ ) + 4 m VT=0.161 \frac{V}{-S \ln (1-\bar{\alpha})+4 m V}T=0.161−Sln(1−aˉ)+4mVV
Among them, mmm is the air sound absorption coefficient,SSS represents the total surface area of the room. α ˉ \bar{\alpha}aˉ represents the average sound absorption coefficient, which can be calculated using the following formula:
α ˉ = 1 S ∑ S k α k \bar{\alpha}=\frac{1}{S} \sum_{} S_{k} \alpha_{ k}aˉ=S1∑Skak
Ering's formula is similar in form to Sabine's formula, but Ering's formula modifies the absorbing term into a logarithmic form. The units and variables of the formula are the same as those defined in the Sabine formula. When the sound absorption coefficient is small, the logarithmic term can be expanded as follows: − ln ( 1 − α ˉ ) =
α ˉ + α ˉ 2 2 + α ˉ 3 3 + ⋯ -\ln (1-\bar{\alpha})=\bar{\alpha}+\frac{\bar{\alpha}^{2}}{2}+\frac{\ bar{\alpha}^{3}}{3}+\cdots−ln(1−aˉ)=aˉ+2aˉ2+3aˉ3+⋯
Ignoring the higher-order terms on the right side of the equation, Erin's formula can be simplified into Sabine's formula considering the sound absorption effect of air. It can be seen from the above derivation that when the sound absorption coefficient is small, the calculation results of Sabine's formula and Erin's formula are close. Therefore, Erin's formula is often used to calculate the reverberation time of rooms with large sound absorption coefficients (such as studios, recording studios).
The reverberation time calculation formulas introduced above are based on the theory of statistical acoustics. However, in actual rooms, the application of these formulas is often subject to the following restrictions: (1) The sound field of the actual room is often inhomogeneous, which does not meet the theoretical assumptions of the diffuse sound field . Even in a hall with professional acoustic design, its sound field is difficult to reach the ideal diffuse sound field standard. (2) The shape of the actual room and the objects and people in the room are very complex and may change. It is difficult to calculate the reverberation time based on the reverberation time calculation formula based on statistical acoustics. Therefore, an acoustic measurement of reverberation time is required. The ISO 3382-1 standard specifies two methods for measuring room reverberation time: interrupted sound source method and impulse response inverse integration method.
The interrupted sound source method is the most intuitive measurement method. The interrupted sound source method uses broadband white noise or pseudo-random noise to input the dodecahedron speaker system as the sound source, so that the sound pressure level at the standard microphone is greater than the background noise by at least 45dB, then turn off the sound source, and record the energy decay curve (Energy Decay Curve, EDC). The decay rate is obtained by linear fitting EDC, and then the reverberation time is calculated. In order to reduce the random error of measurement, the interrupted sound source method needs to take multi-point measurements in the room for averaging, and multiple measurements are required for averaging at each point. Figure 2 shows a schematic diagram of reverberation time measurement using the interrupted source method in a real room.
Impulse response inverse integration method [8] first needs to measure the room impulse response. The room impulse response can be obtained by inputting a broadband signal such as white noise or frequency sweep as an excitation signal at the sound source, receiving the excitation signal propagated through the room at the microphone, and then deconvolving the excitation signal with the signal received at the microphone. , the measurement method of the room impulse response will be introduced in detail in a future article. The acoustic energy at any time can be obtained by inversely integrating the room impulse response with the following formula:
E [ s 2 ( t ) ] = ∫ − ∞ 0 h 2 ( t − τ ) d τ E\left[s^{2} (t)\right]=\int_{-\infty}^{0} h^{2}(t-\tau) d \tauE[s2(t)]=∫−∞0h2(t−t ) d t
The sound energy at each moment can be calculated through the formula to obtain EDC. Similar to the interrupted sound source method, the decay rate can be obtained by linear fitting EDC, and then the reverberation time can be calculated. The impulse response inverse integration method also requires repeated measurements at multiple points, but it has higher measurement accuracy than the interrupted sound source method.
However, due to limitations of recording equipment, recording environment, and measurement methods, some room impulse responses contain a high noise floor. When performing reverse integration, these background noises will change the EDC, thereby affecting the effect of linear fitting and causing deviations in the calculation of reverberation time. Therefore, the calculation of room reverberation time using the nonlinear fitting method [9] is one of the most commonly used methods in current research. The method uses the following theoretical model of the noisy room impulse response:
h ( t ) = ∑ i = 1 NA ie − τ i ( t − t 0 ) sin [ ω i ( t − t 0 ) + ϕ i ] A nn ( t ) h(t)=\sum_{i=1}^{N} A_{i} \mathrm{e}^{-\tau_{i}\left(t-t_{0}\right)} \sin \left[\omega_{i}\left(t-t_{0}\right)+\phi_{i}\right] A_{\mathrm{n}} n(t)h(t)=i=1∑NAie− ti(t−t0)sin[ ohi(t−t0)+ϕi]Ann(t)
This method decomposes the room impulse response into a superposition of individual modes. Among them, respectively represent the initial amplitude, decay rate, angular frequency and initial phase of the room impulse response in the first mode; represent the unit white noise signal, and represent the amplitude of the noise signal. In this method, the theoretical model of room impulse response with noise is used for nonlinear fitting to estimate the total attenuation rate. The reverberation time is calculated using the formula:
T 60 = − 1 τ ln ( 1 0 − 3 ) ≈ 6.908 τ T_{60}=-\frac{1}{\tau} \ln \left(10^{-3 }\right) \approx \frac{6.908}{\tau}T60=−t1ln(10−3)≈t6.908
Key words:
Room reverberation time, reverberation time calculation and measurement
References:
[1] KUTTRUFF H. Room Acoustics[M/OL]. 6th edition. Boca Raton: CRC Press, 2016. DOI: 10.1201/9781315372150. [2]
Dai Genhua. Subjective evaluation of hall sound quality and optimal design parameters[ J]. Acoustic Technology, 1988(02): 37-43+47.
[3] DAVIS J P. Practical Stereo Reverberation for Studio Recording[J]. Journal of the Audio Engineering Society, 1962, 10(2): 114- 118.
[4] KNECHT HA, NELSON PB, WHITELAW GM, etc. Background noise levels and reverberation times in unoccupied classrooms[J]. 2002. [5]
YACULLO WS, HAWKINS D B. Speech recognition in noise and reverberation by school- age children[J]. Audiology, 1987, 26(4): 235-246.
[6] Wang Tao. Hall Sound Quality Design and Subjective Evaluation Research[J]. 2008.
[7] EYRING C F. Reverberation time in “dead” rooms[J]. The Journal of the Acoustical Society of America, 1930, 1(2A): 217-241.
[8] SCHROEDER M R. New method of measuring reverberation time[J]. The Journal of the Acoustical Society of America, 1965, 37(6): 1187-1188.
[9] ANTSALO P, MAKIVIRTA A, VALIMAKI V, 等. Estimation of modal decay parameters from noisy response measurements[C]//Audio Engineering Society Convention 110. Audio Engineering Society, 2001.