How to Calculate Room Acoustic Parameters

With the release of SAFE version 0.3.1 multiple room acoustic parameters have been added including T60, T30, T20, EDT, C50, C80, D50 and Ts. I want to share these algorithms and explain what the parameters mean from a perceptive point of view.

Kasper Kiis

By Kasper Kiis Jensen

The picture above is from Collected papers on acoustics by Wallace Clement Sabine who is said to be the father of architectural acoustics. He developed what is today called Sabine’s formula which is the relation between reverberation time (T60), volume (V) and equivalent absorption area (A):

\[T_{60} = \frac{0.161V}{A} \quad [s]\]

Since 1922 architectural acoustics have evolved and now there are tons of different parameters which can be calculated from a measured impulse response. Especially for concert halls. This blog post will go through many of the different parameters, explaining what they mean and at the end of the post, you can download all the algorithms in Python code.

Reverberation Parameters

Reverberation parameters relate to how much echo there is in a room. This is also called reverberance. This blog post will explain the two most used parameters, Reverberation Time and Early Decay Time.

Reverberation Time

Reverberation Time (T) is the time it takes for the sound pressure level to decay 60 dB. There are three reverberation times T60T30, and T20. They are all valid and applicable at different signal-to-noise ratios. 

T60 is the time it takes for the level to decay from -5 dB to -65 dB.

T30 is the time it takes for the level to decay from -5 dB to -35 dB multiplied by 2.

T20 is the time it takes for the level to decay from -5 dB to -25 dB multiplied by 3.

The most used is T30 as T60 can be very hard to achieve. 

Early Decay Time

Early Decay Time (EDT) is the time it takes for the sound pressure level from 0 dB to decay 10 dB, and is said to be the perceived reverberance. This means that T60, T30 and T20 are more accurate in terms of the correct reverberation time but EDT is the reverberation time we actually perceive.

Clarity Parameters

Clarity parameters relate to how well you can understand speech or music in a room, dependent on reflections from the room. The three parameters explained in this blog post are clarity, definition and centre time which is defined in ISO3382-1.

Clarity

Clarity (C50, C80) describes the relationship between the energy in early sound reflections with late sound reflections. They separate between early and late at 50 ms or 80 ms dependent if the source of the energy is speech or music.  

\[C_{80} = 10log_{10} \frac{\displaystyle \int_0^{80 \textit{ ms}} p^2(t)~dt}{\displaystyle \int_{80 \textit{ ms}}^\infty p^2(t)~dt} \quad [dB]\]
\[C_{50} = 10log_{10} \frac{\displaystyle \int_0^{50 \textit{ ms}} p^2(t)~dt}{\displaystyle \int_{50 \textit{ ms}}^\infty p^2(t)~dt} \quad [dB]\]

The higher a value the better clarity. A good example is listening to a PA system in large rooms such as airports where large delays between the reflections and direct sound makes speech very unclear meaning a low clarity. 

Definition

Definition (D50) describes the energy in early sound reflections relative to all reflections. 

\[D_{50} = \frac{\displaystyle \int_0^{50 \textit{ ms}} p^2(t)~dt}{\displaystyle \int_{0}^\infty p^2(t)~dt} \quad [\cdot]\]

Definition is directly related to clarity and it is therefore not necessary to calculate both parameters.

Centre Time

Centre Time (Ts) is the centre of gravity of the squared impulse response and is the last of the clarity parameters.  

\[T_s = \frac{\displaystyle \int_0^\infty tp^2(t)~dt}{\displaystyle \int_0^\infty p^2(t)~dt} \quad [s]\]

Centre Time is the perceived balance of the room between clarity and reverberation. A low Ts means that a room is very clear while a large Ts means the room is very reverberant.

Sound Level Parameters

The sound level parameter Sound Strength describes the perceived amplification of sound. As rooms can amplify the sound of a sound source such as a loudspeaker or speech. 

Sound Strength

Sound Strength (G) states the sound source energy in a room relative to the sound source energy in free field.

\[G = 10log_{10}\frac{ \displaystyle \int_0^\infty p^2(t)~dt}{\displaystyle \int_0^\infty p_{10}^2(t)~dt}  \quad [dB]\]

A positive Sound Strength means the overall sound level of the room is subjectively higher, with a just noticeable difference of 1 dB. This is especially true in small rooms where Sound strength is more perceived than in large rooms which becomes more reverberant.

Spatial Parameters

Spatial parameters describe the perceived spaciousness or envelopment a room can have on the sound. It is measured using different microphones and equipment than all the other parameters.

Lateral Energy Measures

Lateral Energy Fraction (LF) can be divided into early and late energy measures and are a description of how much energy there is in the side reflections relative to all reflections. 

\[LF_{early} = \frac{\displaystyle \int_{5 \textit{ ms}}^{80 \textit{ ms}} p_L^2(t)~dt}{\displaystyle \int_0^{80 \textit{ ms}} p^2(t)~dt} \quad [\cdot]\]

Early Lateral Energy Measure is understood as the perceived width of the source and is measured between 0 ms and 80 ms.

\[LF_{late} = 10log_{10}\left[\frac{\displaystyle \int_{80 \textit{ ms}}^{\infty} p_L^2(t)~dt}{\displaystyle \int_0^{\infty} p^{2}_{10}(t)~dt}\right] \quad [dB]\]

Late Lateral Energy Fraction takes the entire impulse response into account. The perception is therefore not the width but the envelopment of the room.

Inter-Aural Cross Correlation

Inter-Aural Cross Correlation (IACF) is measured binaurally with a dummy head and correlates well with the spatial impression of a room.

\[IACF = \frac{\displaystyle \int_{t_1}^{t_2}p_l(t)p_r(t+\tau )~dt}{\sqrt{\displaystyle \int_{t_1}^{t_2}p_l^2(t)~dt \cdot \displaystyle \int_{t_1}^{t_2}p_r^2(t)~dt}} \quad [\cdot]\]

Inter-Aural Cross Correlation is not a parameter that is used widely, but binaural measurements are used more and more. 

How To Get The Algorithms

To download the Python implementations of the room acoustic parameters, insert your contact information below and you will receive an email with the code. 

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