Game engine based auralization of airborne sound insulation

Game engine based auralization of airborne sound insulation

J

F

June, 2018

Game engine based auralization of airborne sound insulation

is a project done in the course Master’s Thesis in Engineering Physics, 30.0 ECTS at the Department of Physics, Umeå University, in collaboration with Tyréns AB.

Supervisor: Rikard Öqvist, Acoustics department, Tyréns AB.

Examiner: Krister Wiklund, Department of Physics, Umeå University.

especially when the single number ratings are interpreted by others than experienced acousticians. When developing infrastructure, tools for decision making needs to address visual and aural perception. Visual perception can be addressed using game engines and this has enabled the establishment of tools for visu- alizations of planned constructions in virtual reality. Audio engines accounting for sound propagation in the game engine environment are steadily developing and have recently been made available. The aim of this project is to simulate airborne sound insulation by extending the support of recently developed audio engines directed towards virtual reality applications.

The case studied was airborne sound insulation between two adjacent rooms in a building, the sound transmitted to the receiving room through the building structure resulting from sound pressure exciting the structural elements in the adjacent source room into vibration. The receiving room composed modelled space in the game engine Unreal Engine and Steam Audio was the considered audio engine. Sound trans- mission was modelled by filtering based on calculations of transmission loss via direct and flanking paths using the model included in the standard EN 12354-1.

It was verified that the filtering technique for modelling sound transmission reproduced attenuations in correspondence with the predicted transmission loss. Methodology was established to quantify the quality of the audio engine room acoustics simulations. A room acoustics simulation was evaluated by comparing the reverberation time derived from simulation with theoretical predictions and the simulated reverber- ation time showed fair agreement with Eyring’s formula above its frequency threshold. The quality of the simulation of airborne sound insulation was evaluated relating the sound field in simulation to insula- tion classification by the standardized level difference. The spectrum of the simulated standardized level difference was compared with the corresponding sound transmission calculation for a modelled scenario.

The simulated data displayed noticeable deviations from the transmission calculation, caused by the au- dio engine room acoustics simulation. However, the simulated data exhibited cancellation of favourable and unfavourable deviations from the transmission calculation resulting in a mean difference across the spectrum below the just noticeable difference of about 1 dB. Single number ratings was compared and the simulated single number rating was within the standard deviation of how the transmission model calcu- lates predictions for a corresponding practical scenario measured in situ. Thus, the simulated data shows potential and comparisons between simulated data, established room acoustics simulation software and in situ measurements should further be made to deduce whether the deviations entails defects in the airborne sound insulation prediction or is an error imposed by the audio engine room acoustics simulation.

Keywords: Auralization, Building acoustics, Airborne sound insulation, EN 12354-1, Room acoustics, Audio engines for virtual reality, Game engine.

First of all, I would like to address the exceptional effort by my supervisor Rikard, thank you for your support, your enthusiasm and most importantly, for always broadening my perspective. Rikard, your con- tribution stretches far beyond this Master’s thesis project.

I would also like to express my gratitude towards Tyréns AB enabling me to perform this project, providing equipment, premises and trips, a great working environment and I have been fortunate to be surrounded with pleasant people on daily basis, this acknowledgement is dedicated to you as well. Further, I would like show my appreciation towards all of you from the Tyréns offices at Umeå, Luleå and Stockholm taking your time guiding me and showing interest. Arne, thank you for your support from the beginning of this project and for all helpful discussions providing me with direction. Philip, your time, interest and expertise in room acoustics have been significant.

One thing that I particularly appreciate about Umeå University and would like to acknowledge is the open environment allowing for direct interaction between students and teachers. There have been plenty of influential teachers at Umeå University during my engineering physics studies that deserves recognition.

In connection to this project, I would like to address the efforts by you Krister, your support towards the students is major and your guidance in the process of my thesis project has been valuable. I would as well like to thank you Petter for taking your time and supporting me in the writing process.

This acknowledgement would definitely be incomplete without addressing the great people I have had a chance to be surrounded with during these five years of engineering physics, you that I have had a chance to work with every day as well as having strict lunch hours with! Viktor, you got me through this and you have made everything bearable. Carl, among many things, I would not have managed without the ruthless answers to any question! Karl-Johan, your energy (and expertise in denim) have been significant. I would also like to address: You who was always lightening the mood, when awake and when not, it would have been less fun times without you! You who was wondering how I was reasoning and gave me the nickname that stuck. This is also to the most efficient person ever to proceed an engineering physics education, safe!

To the champ with the neon sign, I would like to get a chance to claim the title some day. Additionally, this definitely calls for the glass version! I have missed plenty, but I am glad you are too many to announce here.

I must stress that the endless support from my family deserves a proper acknowledgement, because your support has been crucial. Finally, I would like to dedicate this to my friends, for instance, always ready for a deg and always ready for a cruise when visiting, as well as to you who I almost never seem to run into except at Friday lunch.

Jimmy Forsman, Umeå, Sweden, 11 June, 2018.

1 Introduction 1

1.1 Background . . . 1

1.2 Aim . . . 2

1.3 Goals . . . 2

1.4 Disposition . . . 2

2 Theory 4 2.1 Acoustics . . . 4

2.1.1 Plane waves . . . 4

2.1.2 Energy transport . . . 5

2.1.3 Sound pressure level . . . 5

2.1.4 Standardized frequency bands . . . 6

2.2 Sound modelling in rooms . . . 6

2.2.1 Surface absorption . . . 6

2.2.2 Scattering . . . 7

2.2.3 Geometrical acoustics . . . 8

2.2.4 Diffuse fields . . . 8

2.2.5 Theoretical reverberation time prediction . . . 9

2.3 Sound transmission . . . 10

2.3.1 Airborne sound insulation . . . 10

2.3.2 The standard EN 12354-1 and flanking transmission . . . 12

2.4 Signal theory for auralization . . . 15

2.4.1 Fourier transform . . . 15

2.4.2 Convolution, impulse response and transfer function . . . 16

2.4.3 Retrieving an impulse response – Dirac delta function . . . 16

2.5 Reverberation time derivation . . . 17

2.6 Room impulse response . . . 18

2.7 The concept of head related transfer functions . . . 19

2.8 Ambisonics . . . 19

3 Method 21 3.1 Modelling approach . . . 21

3.2 Transmission loss based filtering . . . 24

3.2.1 Performance parameter input data . . . 24

3.2.2 Designing wall-filters . . . 26

3.2.3 Wall-filtering . . . 28

3.3 Receiving room simulation . . . 28

3.3.1 Audio engine . . . 29

3.4 Evaluation . . . 29

3.4.1 Evaluation setup . . . 30

3.4.2 Filtered secondary source signals . . . 33

3.4.3 Reverberation time . . . 33 3.4.4 Virtual sound insulation measurement . . . 35

4 Results 39

4.1 Transmission loss based filtering . . . 40 4.2 Reverberation time . . . 41 4.3 Virtual sound insulation measurement . . . 42

5 Discussion 47

5.1 Transmission loss based filtering . . . 47 5.2 Reverberation time . . . 47 5.3 Virtual sound insulation measurement . . . 48

6 Conclusions & outlook 51

6.1 Future work . . . 52

Bibliography 54

Introduction

1.1 Background

The current population growth demands a high rate of developing infrastructure. The United Nations predicts that 66 percent of the world’s population will reside in urban areas by 2050, compared to the 54 percent in 2014. The estimated population growth then corresponds to adding 2.5 billion people to urban areas in the time span 2014 to 2050 [1]. A high rate of urbanization requires agile decision making from involved engineers, clients and contractors. Availability of efficient support for decision making is desirable to make proper judgements early in the building process for preventing the rapid development from exerting unnecessary marks on the environment by in hindsight revising built constructions. Support for making the technical decisions is crucial, however, the development would not be sustainable if the inhabitants of the constructed environments are neglected. Therefore, tools for rational decision making are required to manage how planned constructions may be perceived.

The usage of virtual reality systems are today applied in various disciplines in science and engineering, where a major aspect of its usefulness is the ability to address human perception. Virtual reality applications commonly aim to manage the human visual system. A broader modality is strived for to improve the ability of managing perception and can be achieved through consolidating the visual stimulation with aural stimulation. Virtual reality systems often relies on game technology where sufficient visual stimulation for many purposes can be achieved utilizing techniques of computer graphics, thus allows for comprehensive visualizations. To broaden modality of virtual reality systems the visualization is to be supplemented by its audio analogue, auralization. Auralization is a term introduced in a paper from 1993 by Kleiner et al. [2]

and can be considered to render audible sound fields for simulating the acoustics of modelled space. The integration of auralization into virtual reality systems based on established game engines have not been conducted to the same extent due to the relatively less developed support of the game engines in simulating sound propagating in the virtual environment.

Tyréns AB have developed a platform for visualization aimed towards sustainable development in con- nection to civil engineering [3]. The visualization platform can be used in the design process when devel- oping infrastructure and provides an impression of how a planned project may be visually perceived prior to construction. The visualization platform—TyrEngine—is based on the game engine Unreal Engine which allows for immersing an observer by means of virtual reality into planned constructions for real-time visu- alization of three dimensional environments. TyrEngine is hence based on computer game technology and the well developed graphical support of the game engine have in various projects proven sufficient for ad- dressing the human visual system. However, TyrEngine has currently no support for addressing the human auditory system. Support for addressing aural perception would enable for communicating the effects of acoustic design. One reason for the lacking support is then due to the much less developed features of the game engine for taking into account the relevant physical properties of sound propagating in the modelled environment.

Recently, companies such as Valve and Google have devoted attention to the development of audio engines aimed towards integration with game engines providing more detailed modelling of how sound interacts with the virtual environment. During the past year, the newly developed audio engines have been

made available [4, 5]. It is of interest to investigate how the recently developed audio engines can be util- ized for civil engineering applications performing game engine based auralization simulating perceivable effects of different acoustic solutions. This Master’s thesis will focus on taking the initial steps for broad- ening modality of game engine based visualization platforms combining visualization with auralization by utilizing the inherent features, as well as extending the support, of recently developed audio engines.

Often, acoustical engineers describes the performance of a construction in isolating against airborne sound by providing single numbers ratings and graphs indicating insulation properties in frequency bands.

Data in terms of single number ratings and graphs are far from an exhaustive description of the subjective event that clearly illustrates perceptual consequences of planned acoustic design, hardly an effective tool facilitating prudent decision making. Thus, single number ratings as a decision basis when interpreted by others than experienced acousticians imposes risks of sub-optimal constructions resulting in unsustainable development regarding human health.

In this thesis, development towards vivid descriptors of sound insulation to function as adequate de- cision making tools will be carried out by studying a case involving—auralization of airborne sound insu- lation—the modelling and simulation of the sound field inside the receiving room as an effect of airborne sound transmitted from the adjacent source room. To entail possibilities of supplementing high quality visualizations by auralization, simulations will be confined to the game engine environment, specifically, Unreal Engine [6] using Valve’s Steam Audio [7] as a basis for modelling sound propagation in the virtual environment. The aim in a future perspective is to enable for physically accurate virtual reality presenta- tions of acoustic design proceeding from the versatile game engine environment. The first step is to proceed from algorithms developed for auralization of airborne sound insulation [8, 9], performing modifications adapting the algorithms to the game engine environment and further performing objective evaluations of this approach for building acoustical auralization. Objective verifications of game engine based building acoustical auralization will benefit the development of tools for clearly describing the influence of acous- tical design in civil engineering. Thereby simplifying decision making for engineers, clients as well as contractors when striving towards creating sustainable, acoustical, environments. The initial steps in the development process will be taken during the course of this thesis.

1.2 Aim

The aim of this project is to simulate airborne sound insulation by extending the support of recently de- veloped audio engines directed towards virtual reality applications.

1.3 Goals

The following goals are specified:

i. Perform and evaluate a game engine based room acoustics simulation.

ii. Simulate airborne sound transmission between two adjacent rooms in a building using established models dedicated to sound transmission calculations.

iii. Evaluate the quality of the simulation using standardized building acoustical methodology to relate the simulated sound field to numerical descriptors of sound insulation.

iv. Deliver methodology to be used with recently developed audio engines for performing and evaluating game engine based auralization.

1.4 Disposition

Acoustics is briefly introduced in Chapter 2 proceeding from wave theory. Central concepts for auralization are then addressed after dealing with common models in room and building acoustics. The modelling of sound transmission as well as methodology for performing and evaluating game engine based auralization is the topic of Chapter 3. Results from various evaluations are provided in Chapter 4, analysed in Chapter

5 and concluding remarks from the analysis in connection to the specified goals are presented in Chapter 6. Chapter 6 is then finished off by emphasizing where to proceed with further research from what is established in this thesis.

Theory

2.1 Acoustics

Acoustics in terms of for instance airborne or structure-borne sound concerns waves in air or solid media.

Here, we will introduce central acoustical concepts proceeding from wave theory.

Particles in a sound wave follows a space and and time dependent displacement vector, s, describing relative displacement from particle equilibrium with particle velocity, v, given by the derivative of s with respect to time as

v = ∂ s

∂ t. (1)

For airborne and structure-borne sound the particles then concerns air molecules and atoms in a crystal lattice. The related density and pressure fluctuations, ρ and p, of the medium which the sound wave propagates through can be expressed

ρ = ρ_tot− ρ0, (2)

p= p_tot− P₀, (3)

where the totals indicates the effective quantities of the medium as a result of the displacements from the equilibrium density ρ0and pressure P₀, respectively. The sound pressure, p, is the main quantity of interest and it is the sound pressure that relates sound waves to human hearing [10]. We will mainly deal with sound waves in fluid media for now and focus on airborne sound. The pressure fluctuations due to sound waves is assumed to fulfill the wave equation

c²∇²p−∂²p

∂ t² = 0, (4)

where c is the speed of sound and we stick with physics notations such that ∇²denotes the Laplacian. The sound speed is the phase speed of the sound wave in the medium, that is, an observer following the sound wave at speed c would experience a constant phase and see no change in the wave pattern, thus, phase speed [11].

2.1.1 Plane waves

We will introduce necessary concepts of wave theory related to acoustics and therefore we consider a solution of the wave equation as a plane harmonic wave

p(r,t) = ˆpe^{i(ωt−k·r)}, (5)

where ˆpis the, generally complex, pressure amplitude, ω is the angular frequency, k is the wave number vector and r denotes the position vector. Note that when dealing with actual physical quantities we consider only the real part. Now, consider the position vector r for a Cartesian coordinate system where r ∈ R³, we

then have that r = (x, y, z) = xˆx + yˆy + zˆz and ˆx, ˆy, ˆz denotes the unit vectors in each coordinate direction.

Further, the wave number vector k is related to the wavelength λ and the angular frequency ω is related to the frequency f as

k =2π

λ ˆn, (6)

ω = 2π f , (7)

where ˆn denotes the unit vector normal to a plane perpendicular to the direction of propagation. We can now introduce the period, T_P, of the wave as

TP=1

f. (8)

2.1.2 Energy transport

Two governing quantities for energy transport by sound waves are the sound power W and the sound intensity I, quantities that commonly are described by time averages. The sound power describes the flow of sound energy through a surface S per unit time and the sound intensity describes mean energy flow as the transmitted sound power through a reference surface of area 1 m², a surface laying in a plane perpendicular to the direction of which the sound wave propagates [11, pp. 61-62]. The sound intensity and power are given by

I = 1 t_A

ˆ _tA

0

pvdt, (9)

W =

˛

S

I · dS, (10)

where t_Ais a sufficiently long measuring time [12, pp. 15]. Through the characteristic impedance of plane waves [10, pp. 14], Z₀, given by

Z0= p

v = ρ0c, (11)

where v here denotes the component of the particle velocity, the sound intensity and power for plane waves may be expressed

I= kIk = p˜²

ρ0c, (12)

W =

˛

S

˜ p²

ρ0cdS, (13)

where ˜pis the root mean square sound pressure defined by

˜ p=

s 1 tA

ˆ _tA

0

p²(r,t)dt. (14)

2.1.3 Sound pressure level

We move on by introducing the sound pressure level, L, given in decibel (dB) scale as L= 10 log₁₀p˜²

p²₀, (15)

where the constant p₀denotes the reference sound pressure, for air, taken as p₀= 20 µPa which is an approximate lower limit for audible pressure fluctuations by human hearing in midfrequencies [10]. For practical applications considering the human perception of sound, the decibel scale is suitable to discuss with a resolution of 1 dB corresponding to the just noticeable difference (JND) in audible sounds [10,

pp. 18]. If the total level is desired for a scenario with several signals present, each signal corresponding to a certain sound pressure, we are required to calculate the root mean square of the total sound pressure according to Eq. (15). It is therefore vital to consider whether the signals are coherent or not. The total pressure is obtained by simple summation of the included sound pressures and the total pressure is squared when evaluating the corresponding root mean squared sound pressure according to Eq. (14). For illustrative purposes, consider the squared sum of two signals having sound pressures p₁and p₂as

(p₁+ p₂)²= p²₁+ 2p1p₂+ p²₂, (16) for coherent signals with equal frequency and a certain phase relation, each term in Eq. (16) is contributing.

However, for the case of incoherent signals of various frequencies, the left-hand side of Eq. (16) is equal to adding squared sound pressures because the coupled term cancels. The total energy density wtot, and further the total level, can then for Ns incoherent signals be expressed validly in terms of summation of squared root mean squared sound pressures

w_tot=

Ns

∑

w_i=

Ns

i=1

∑

˜ p²_i

ρ0c², (17)

L= 10 log₁₀

Ns

∑

i=1

p˜²_i

p²₀ , (18)

as opposed to the case of coherent signals as illustrated by the simple arithmetic example in Eq. (16) [10, pp. 20-21].

2.1.4 Standardized frequency bands

It is common in acoustics to divide the spectrum into frequency bands, octave bands and particularly fractional octave bands are frequently used. We consider one-third octave bands and their logarithmic composition by proceeding from a single band where f_ldenotes the lower edge frequency. The upper edge frequency, f_u, width, ∆ f , and center frequency, fc, are then given by

f_u= 2^1/3f_l, (19a)

∆ f = f_u− f_l, (19b)

f_c=p

f_lf_u, (19c)

f_c+1= 2^1/3f_c, (19d)

where f_c+1is the center frequency for the above sequent band [10, pp. 21-22].

2.2 Sound modelling in rooms

Simulating airborne sound insulation requires modelling of the sound field inside enclosures. To simulate sound fields inside rooms, as well as sound transmission, we here turn to the models commonly used in room and building acoustics. Not seldom, models in room acoustics are based on geometrical acoustics for providing quite simple, or rather, efficient descriptions of sound fields inside rooms when the purpose is to address human perception. In addition, the approximation of diffuse fields is commonly used in the two areas that is building and room acoustics. Diffuse fields are introduced in Section 2.2.4 after some essential concepts to address sound propagation inside rooms.

2.2.1 Surface absorption

For defining the reduction in energy of a sound wave propagating in space interacting with a surface we will consider a plane wave incident on an assumed infinitely large smooth surface for which the sound wave is specularlyreflected following Snell’s law [13, pp. 15]. Snell’s law being that the angle of incidence equals

the reflection angle as θi= θr= θ . Let the infinitely large surface lie in the y-z plane located at x = 0 such that the, complex, reflection factor ˆR[12, pp. 36] can be expressed in terms of the incident wave p_iand the reflected wave p_ras

p_i(x, y,t) = ˆpei(ωt−kx cos θ −ky sin θ ), (20a) p_r(x, y,t) = ˆRpeˆ i(ωt+kx cos θ −ky sin θ ), (20b) where the angle of incidence θ is defined with θ = 0 corresponding to normal incidence to the surface and θ ∈ [0, π/2]. We can then express the reflection factor in terms of the ratio between the reflected and incident wave and further through the wall impedance [10, pp. 36], Z, as

Rˆ= pr

p_i =Zcos θ − Z0

Zcos θ + Z0

, (21)

where Z is defined at the surface as the ratio between the sound pressure p and vn, given by Z= p

v_n  

_surface, (22)

where v_ndenotes the component of the particle velocity vector normal to the surface [12, pp. 37]. We then define the absorption coefficient, α, as

α = 1 − | ˆR|². (23)

Further, when addressing absorption or the absorption coefficient α in the context of diffuse fields we will not be referring to the absorption coefficient dependent on a specific angle of incidence, instead we will refer to the diffuse or averaged absorption coefficient, αdiffuse, which is the absorption coefficient for random sound incidence [12, pp. 55] given by

αdiffuse= ˆ _{π /2}

0

α (θ ) sin(2θ )dθ . (24)

2.2.2 Scattering

Sound interacting with a surface containing irregularities is normally quantified by the scattering coeffi- cient, s, defined in the standard ISO 17497-1 [14]. Before we go on stating the definition of the scattering coefficient we briefly go though the necessity of quantifying sound scattering. The surface which the sound wave interacts with is often not completely smooth as it was assumed when defining surface absorption in Section 2.2.1. In practice, the surface which the sound wave interacts with frequently contains irregularit- ies on the same order as the wavelength of the incoming sound. For the two cases when the wavelength is much greater than or much less than the dimensions of the irregularities, the sound wave can be assumed to be specularly reflected off of the surface or off of the surface irregularities. When the dimensions of the surface irregularities and the wavelength is of the same order, we will experience a diffusely reflected wave where part of the sound wave is specularly reflected and part of the sound wave is scattered, which effectively spreads the sound wave in a multitude of different directions [13, pp. 16]. Imagine that we are confined to a small room enclosed by low absorbing surfaces containing irregularities scattering the in- coming sound, we realise that the sound field in such an enclosure would entail a random, or rather, evenly distributed field over time. The distribution of the scattered sound is described exactly by an associated directivity pattern [11, pp. 138]. In room acoustics for instance, such an exact representation does not favour many of the modelling purposes and we instead define the scattering coefficient according to ISO 17497-1 [14] simply as

s= 1 −E_r,specular Er,tot

, (25)

where E_r,specularand E_r,totdenotes the specularly reflected acoustic energy and the total reflected acoustic energy, respectively. Methods to be used for determining such random sound incidence scattering coeffi- cients as is Eq. (25) are further standardized in ISO 17497-1 [14].

2.2.3 Geometrical acoustics

The description of the sound field by geometrical acoustics is similar to the concept of for instance geo- metrical optics. In geometrical acoustics the sound field is described by energy carrying particles rather than waves. The particles are addressed as rays having direction and specifically, a ray corresponds to a spherical wave with opening angle Ω approaching zero: Ω → 0. Following from the spherical wave rep- resentation, the sound intensity I obeys the inverse-square law and decreases with distance from the origin ras

I ∝ 1

r². (26)

The particle description of rays neglects certain wave phenomena since the particles, or rays, are to be considered incoherent. Important to note is that the approximation of geometrical acoustics is confined to broadband signals, above the so called Schroeder cut-off frequency and where the room dimensions are much greater than the wavelengths of the sound [10, pp. 58-59].

2.2.4 Diffuse fields

For measurements as well as calculations carried out in the field of building acoustics, the focus often revolves around determining the squared sound pressure averaged in time and space. Another measurement of interest regards quantifying the ability of an enclosure in sustaining energy when excited by a sound source, reverberation, and is quantified by the reverberation time, T . Consider a source feeding sound power into an enclosure, for instance a closed room confined only by the room boundaries. The moment the source is turned off the sound will not stop instantly, rather it will steadily decay during a period of time determined by the energy absorption properties of the room boundaries. Reverberation is precisely sound decay and the reverberation time is a specific decay time commonly denoting the time for a level decrease of 60 dB. A common approach predicting for instance reverberation time is to describe the sound field using statistical models under the assumption that diffuse field conditions are present. The frequency range when statistical modelling becomes applicable inside enclosures is when low frequency behaviour such as room modes are no longer dominating. A lower limit for statistical modelling is given by the Schroeder cut-off frequency, f_S[11, pp. 116], expressed in terms of room volume V and reverberation time T as

f_S= 2000 rT

V. (27)

Above the Schroeder cut-off frequency, modelling techniques that are common in building and room acous- tics have a tendency to drift away from describing certain wave effects that governs the sound field. The transition into diffuse field modelling is made by describing the sound field by its energy distribution where the energy density is the main acoustic variable [11, pp. 117]. There is a multitude of models used in the field of acoustics where energy serves as the descriptor of the sound field. The important part to note regard- ing the energy based models are their mutual disregard of particular wave phenomena, recall the “sound particle” description used in the approach of geometrical acoustics. Consequently, it should be emphasized that turning to statistical modelling requires assurance that the underlying assumptions are indeed satisfied if moderately accurate results are to be expected. Statistical and diffuse field models favours mainly prac- tical applications and when the aim is to approximate a vast and complex scenario for describing the pure essentials to enable addressing the human auditory system. However, all energy based models does not require that diffuse field conditions are satisfied, examples of such models are the geometrical acoustics based methods of ray tracing and method of images [15, pp. 42].

Finally, it is appropriate to discuss the concept of diffuse sound fields and a rather hand-wavy, but adequate definition is indeed: An ideal diffuse field implies uniform energy density inside the enclosure.

[11, pp. 117]. Unfortunately, a strict definition nor substantial measurements are unavailable for precisely defining what constitutes a diffuse sound field in a practical scenario [15, pp. 42]. It is common to proceed from a theoretical point of view and discussing so called room acoustical parameters and their influence on the varying accuracy of the predictions produced by diffuse field based models. For further insight regarding the simplifications present when dealing with diffuse field conditions, in [11] the author Vigran states two suggestions for definitions which includes essential underlying assumptions. The two suggested definitions are hereby presented as exact quotes from Vigran [11, pp. 117] and are:

– “In a diffuse field the probability of energy transport is the same in all directions and the energy angle of incidence on the room boundaries is random.”

– “A diffuse sound field contains a superposition of an infinite number of plane, progressive waves making all directions of propagation equally probable and their phase relationship are random at all room positions.”

The implication of the first suggested definition has been concerned in Eqs. (24) and (25) when introducing the random sound incidence absorption and scattering coefficients. To further clarify an important implic- ation from the latter suggested definition, the assumption that phase relationship are random results in that interference caused by specific phase relations are neglected in a diffuse field. The simplification imposed is that the sound field consists of a superposition of incoherent plane wave components and because they are assumed incoherent, the total energy of the sound field at a certain position can be described as a sum of the energy of each plane wave component.

The room acoustical parameters taken as a basis for discussing accuracy of diffuse field theory models commonly regards room dimensions, surface absorption, and scattering. Lastly comes a parameter not dealing with the room boundaries and concerns fittings, additional objects in the room scattering the in- coming sound [15, pp. 41]. The strength in considering the sound field as diffuse is that the assumption of uniform directional distribution is rather feasible when considering sound propagating in a closed room containing irregularities [12, pp. 115] and gives rise to fairly simple formulae predicting properties of sound fields in rooms and buildings. The purpose justifies the models, addressing human perception of sound the diffuse field models fixates the bare essentials.

Now, with a slight insight regarding diffuse field conditions, their implications and common room acoustical parameters for addressing the applicability of diffuse field theory based models, it is then appro- priate to introduce prediction models used in the field of room and building acoustics for which plenty of them are derived from diffuse field theory.

2.2.5 Theoretical reverberation time prediction

In addition to that we are above the Schroeder cut-off frequency, general criteria for when approximate diffuse field conditions are present and corresponding formulae may be applicable is that room dimensions are similar, surface absorption of the room boundaries is evenly distributed and the equivalent absorption area, Eq. (33) below, is small. The two reverberation time predictors of interest here, Sabine’s formula and Eyring’s formula, may be derived from classical diffuse field theory [11] and in addition to that approximate diffuse field conditions are fulfilled, the applicability of Sabine’s and Eyring’s formula are dependent on that absorbing material only regards the room boundaries which is the case when the enclosure is free from fittings. The reverberation time formulae are stated for a room of volume V , consisting of N_Broom boundaries, each with associated surface area S_i and absorption coefficient αi, i = 1, 2, . . . , N_B, such that the total surface area S_totis

Stot=

NB

i=1

∑

S_i, (28)

and we express Eyring’s formula as

T= −24 ln 10 c

V

Stotln (1 − ¯α ), (29)

where ¯α denotes the mean absorption coefficient given by α =¯ 1

S_tot

NB

∑

i=1

S_iαi. (30)

Considering the case of low a mean absorption, ¯α  1, and expanding − ln (1 − ¯α ) in Eq. (29) in a Maclaurin series neglecting higher order terms

− ln (1 − ¯α ) = ¯α +α¯² 2 +α¯³

3 + . . .

= { ¯α  1} ≈ ¯α ,

(31)

Eyring’s formula, Eq. (29), can be approximated by T≈24 ln 10

c V

S_totα¯ =24 ln 10 c

V

A, (32)

known as Sabine’s formula where

A= S_totα ,¯ (33)

is the equivalent absorption area of the room. Accounting for air absorption by introducing the air atten- uation coefficient m and letting the time dependent acoustic energy be effected by an additional factor of e^−mct, the corresponding reverberation time formulae, Eyring’s and Sabine’s, may be expressed

T = −24 ln 10 c

V

S_totln (1 − ¯α ) + 4mV, (34)

T =24 ln 10 c

V

A+ 4mV. (35)

However, for small rooms the air absorption term 4mV in Eqs. (34) and (35) can be neglected [12] such that we can stick with Sabine’s and Eyring’s formula in the form of Eq. (29) and Eq. (32).

2.3 Sound transmission

Consider sound transmission between two adjacent rooms, free from fittings, inside a building. The trans- mission process begins with a source emitting sound in one room, the source room. The sound source results in airborne sound exciting the structural elements inside that room and the airborne sound is then converted into structural waves propagating through the connected constructional elements composing the building. The structural elements begins radiating airborne sound upon excitation, airborne sound that fur- ther propagates through space and enables to reach a listener positioned in the adjacent receiving room. At this point, the reader has most certainly thought of a great of amount of additional ways for which sound may be transmitted between the adjacent rooms, for example, out and in through windows interconnected via the building façade, through doors coupling the rooms by corridors running alongside the rooms and so forth. The description above regards airborne sound transmission via coupled structural elements which are made up by the separating element, the main dividing wall, as well as through flanking partitions, such as the floor, the roof and the lateral walls enabling for direct and flanking sound transmission, which is subject to modelling here.

2.3.1 Airborne sound insulation

For a given surface, the transmission coefficient τ is defined as a ratio transmitted sound power Wt over incident sound power W_ias

τ =W_t

W_i. (36)

In the field of building acoustics, the performance of a single constructional element in isolating against airborne sound is commonly quantified by the sound reduction index, R, expressed in decibel as

R= 10 log₁₀1

τ. (37)

Assuming that the sound field in the source room is diffuse, the sound power incident on the element of surface area S can be expressed in terms of sound intensity I_ias

W_i= I_iS=cw_S

4 S= p˜²_S

4ρ0cS, (38)

where w_Sand ˜p_Sdenotes the energy density and the root mean square sound pressure in the source room, respectively. The leftmost equality in Eq. (38) corresponds to the assumption that the sound field is diffuse, the second equality from the left comes from diffuse field theory and the rightmost equality corresponds

to the assumption that the sound field is a superposition of plane waves [11, pp. 118-119]. Assuming also that the sound field in the receiving room is diffuse, an expression from diffuse field theory for the power transmitted through the partition to the receiving room, W_t, in terms of the root mean square sound pressure in the receiving room, ˜p_R, and the equivalent absorption area in the receiving room, A_R[11, pp. 240], is

Wt= p˜²_R

4ρ0cA_R. (39)

Expressing the transmission coefficient τ in terms of Eq. (38) and (39) yields

τ = p˜²_R

˜ p²_S

A_R

S . (40)

Substituting the above relation into Eq. (36), the sound reduction index takes the form R= 20 log₁₀ p˜_S

p˜R

+ 10 log₁₀ S AR

= D + 10 log₁₀ S A_R,

(41)

where D denotes the level difference given by

D= L_S− L_R= 20 log₁₀ p˜_S

˜

p_R, (42)

where L_Sand L_Rare the average sound pressure levels in the source and receiving room and the rightmost equality in Eq. (41) gives the sound reduction index R on the form commonly used in building acoustics.

The assumption of diffuse sound fields in the source and receiving room for which Eq. (41) is valid can be interpreted such that the distance between the sound source and the partition, as well as the distance between the partition and the receiver are greater than the critical distance for each of the two rooms [13].

The critical distance refers to the distance where the sound pressure levels are equal for both the direct and reverberant sound.

When describing a material, a structural element, using the sound reduction index, it is based on meas- urements in a laboratory environment only taking into consideration sound energy transmitted via the actual element. For in situ measurements, sound energy is transmitted over several transmission paths, for example through flanking elements, through windows interconnected via the building façade as well as through ventilation systems and so forth. In such situations the separating element, especially in the case of high sound insulation, is not representable for the actual measured sound pressure level. Therefore, the measured quantity in an actual building is instead referred to by the apparent sound reduction index R⁰[11], determined from measurements by Eq. (41) [16]. The quantity R⁰thus indicates the resulting sound reduc- tion index for a constructional arrangement, all transmission paths considered. For modelling purposes, it is suitable to express R⁰in terms of the apparent transmission coefficient, τ⁰, as

R⁰= 10 log₁₀ 1

τ⁰, (43)

where the difference between τ⁰ and τ is that the former regards all transmission paths in the considered scenario, rather than only a single surface which is the case for the latter [10]. Using the apparent for- mulation, the ratio τ⁰ is expressed according to the standard EN 12354-1 [16] as the total sound power transmitted to the receiving room, W_tot, divided by the sound power incident on the separating element, W_D, such that

τ⁰=Wtot

W_D. (44)

A model accounting for flanking transmission is further covered in Section 2.3.2. Another approach for de- scribing sound transmission between adjacent rooms without including the partition area are the so called in situsound insulation descriptors [17]. The in situ descriptor that we will focus on concerns the standard- ized level difference, D_nT. The reason for devoting our focus to D_nTis that it is a quantity more suitable for

auralization than for instance R⁰, because of its closer relation to how sound is perceived by humans [10].

Finally, D_nT is expressed with reference to the receiving room reverberation time T and is given by D_nT= D + 10 log₁₀ T

T₀, (45)

where T₀is a reference reverberation time that is standardized and commonly set to T₀= 0.5 s for dwellings [11].

2.3.2 The standard EN 12354-1 and flanking transmission

To predict the performance of a certain constructional arrangement in isolating against airborne sound we will use the model included in the standard EN 12354-1 [16]. The EN 12354-1 model is devoted to sound transmission calculations between two adjacent rooms in a building via the direct path, through the separating element, as well as through various flanking paths via a single junction connecting the elements. The considered sound transmission model is based on available data for structural elements as well as junction types. The model delivers R⁰, which further can be converted to D_nT as will be shown later in Section 3.2.1 and R⁰ is delivered expressed in terms of contributions from sound transmitted via different paths. Before presenting the EN 12354-1 model, we define flanking transmission and introduce the conventional nomenclature used when addressing sound transmission by direct and flanking paths.

Consider two adjacent rooms in a building coupled by structural elements and divided by a separating element. Contributions of sound transmission involving the back wall in the adjacent room scenario are neglected in the EN 12354-1 model. An adjacent room scenario showing all structural elements for which sound transmission is accounted for by the EN 12354-1 model is provided in Section 3.2.1, depicted in Fig- ure 4. Neglecting the contributions from the back wall is analogous to only considering sound transmission via a single junction, that is, via first-order junctions. Flanking sound transmission considering first-order junctions regards the energy transport from the source room to the receiving room by the following means:

The transport of energy begins with a sound source exciting constructional element i in the source room into vibration. Part of the vibration energy is then transmitted across one junction to an element j in the receiving room. The sound radiated by element j is the sound transmitted via flanking transmission such that flanking elements are all elements of indices i and j except for the separating element. In the following when introducing the sound transmission model, two major assumptions are present in addition to that only first-order junctions are considered. The two assumptions are, firstly, that the various transmission paths are assumed independent. Secondly, that diffuse field conditions are present in the source and receiving room.

For the case of a separating element and a single flanking element, four different transmission paths are possible as illustrated in Figure 1. Firstly, the direct direct transmission path which is denoted by Dd, capital letters denotes the path in the source room and lower case letters in the receiving room. The flanking paths for the single flanking element scenario then consists of the direct flanking path D f , the flanking direct path Fd and the flanking flanking path F f .

Figure 1 – Illustration of two adjacent rooms indicating direct and flanking transmission paths from the source room (left) to the receiving room (right) via first-order junctions for a scenario where the rooms are coupled by a single flanking element (top) and a separating element (center). Indices i and j corresponds to source and receiving room elements, respectively. Upper case letters corresponds to the path in the source room and lower case letters the receiving room.

Continuing with the discussion around Eq. (43) and Eq. (44) considering the above mentioned transmission paths for the case of n flanking elements allows for expressing the apparent transmission coefficient τ⁰by an addition of sums as

τ⁰= τDd+

n F=1

∑

τFd+

n f

∑

τD f+

n F= f =1

∑

τF f. (46)

We may as well express τ⁰and also R⁰, through Eq. (43), in two additional forms. We begin with the form which can be considered expressing τ⁰ in terms of sound transmission via different paths leading to the various elements in the receiving room and we express τ⁰as

τ⁰= 10^−Rd/10+

n f

∑

10^−Rf/10, with (47a)

10^−Rd/10= τ_Dd+

n F=1

∑

τ_Fd= τ_d, and (47b)

10^−Rf/10= τD f+ τF f = τf. (47c)

In Eq. (47) we have introduced the total direct sound reduction index R_d and the total flanking ele- ment sound reduction index R_f, as well as their corresponding total direct transmission coefficient τdand total flanking element transmission coefficient τf. The quantities R_d and R_f provides sound insulation descriptors via various paths leading to each structural element in the receiving room, back wall excluded.

The quantity R_ddefined in Eq. (47b) is related to the total sound radiated by the separating element in the receiving room and R_f, defined in Eq. (47c), is related to the total sound radiated by flanking element f in the receiving room. In principle, both R_d and R_f takes into account sound incident on the separating element as well as on the flanking elements in the source room and its transmission to the receiving room across at most one junction. However, an assumption in the EN 12354-1 model is that the sound power incident on the source room elements are all approximated by the sound power incident on the separating element, as expressed in Eq. (44). Further, we can also express τ⁰as

τ⁰= 10^−RDd/10+

∑

i, j

10^{−Ri j/10}, with (48a)

10^−RDd/10= τ_Dd, and (48b)

∑

i, j

10^{−Ri j/10}=

n

∑

F=1

τFd+

n

∑

f=1

τD f+

n

∑

F= f =1

τF f, (48c)

by introducing the direct direct sound reduction index R_Dd, defined in Eq. (48b), as well as the flanking sound reduction index R_{i j}related to its corresponding transmission coefficients as shown in Eq. (48c). The flanking sound reduction index R_{i j}is related to the the flanking sound transmission coefficient τi jby

R_{i j}= −10 log₁₀τi j, where (49a)

τ_{i j}=Wi j

W_D = IjSj

I_iS_D, (49b)

and W_{i j}denotes the sound power radiated by flanking element j in the receiving room as a consequence from sound incident on element i in the source room. The ratio τi jis expressed in terms of the sound power incident on a reference area in the source room, chosen according to the standard EN 12354-1 [16] as the area of the separating element S_D. The simplification of using a common reference area when expressing τ_{i j}introduces the assumption that S_i= S_D. However, it is the simplification of a common reference area that allows for expressing the resulting transmission by means of a summation of the transmission coefficients for the present, assumed independent, transmission paths. Inserting τ⁰as expressed by Eq. (48a) into Eq.

(43) we can express the apparent sound reduction index as

R⁰= −10 log₁₀ 10^−RDd/10+

∑

i, j

10^{−Ri j/10}

!

, (50)

where the relation between R_Dd, R_{i j} and their corresponding transmission coefficients are shown in Eq.

(48). If there is any ambiguity about the limits of the sum over R_{i j}, note that this is strictly convention and the sum is presented in Eq. (48c). Now, to provide a clear illustration of the components of the apparent transmission coefficient τ⁰and its components relation to the corresponding sound reduction indices, for the typical case of n = 4 flanking elements, τ⁰can be expressed explicitly as

τ⁰= τd+ τ1+ τ2+ τ3+ τ4

= 10^−Rd/10+ 10^−R1/10+ 10^−R2/10+ 10^−R3/10+ 10^−R4/10, with (51a) τ_d= τ_Dd+ τ_1d+ τ_2d+ τ_3d+ τ_4d

= 10^−Rd/10

= 10^−RDd/10+ 10^−R1d/10+ 10^−R2d/10+ 10^−R3d/10+ 10^−R4d/10, and

(51b)

τj= τD j+ τi j

= 10^−Rj/10

= 10^{−RD j/10}+ 10^{−Ri j/10}, for i = j = 1, 2, 3, 4.

(51c)

What remains is to express the sound transmission and hence, R_{i j}, in terms of properties related to the constructional arrangement. The sound transmission model will not be derived here, rather presented with a brief explanation of the included terms and for a thorough derivation consider Vigran [11, pp. 343-355].

The quantity used by the EN 12354-1 model replacing R_{i j}is the mean value R_{i j}of the sound transmission exchanging both rooms and is on first basis expressed

R_{i j}=R_{i j}+ R_ji

2 =R_i+ R_j

2 + D_{v,i j}+ 10 log₁₀ S_D

pS_jS_i, (52)

where D_{v,i j}denotes the direction averaged velocity level difference given by D_{v,i j}=Dv,i j+ D_{v, ji}

2 . (53)

One remaining obstacle to obtain predictions of flanking sound transmission is that the velocity level differ- ence across the junction must be determined. The difficulty arises because D_{v,i j}is not an invariant quantity related to the junction, but also depends on energy losses inside the receiving element [11, pp. 348]. Seek- ing an invariant quantity for characterizing transmission across a junction, a quantity called the vibration

reduction index is defined [16]. The vibration reduction index is denoted K_{i j}and describes energy trans- mitted across a junction coupling the structural elements i and j and is valid under diffuse field conditions.

From K_{i j}it is possible to obtain the velocity level difference across a junction taking into consideration the energy losses in the elements and K_{i j}is given by

K_{i j}= D_{v,i j}+ 10 log₁₀ l_{i j}

√a_ia_j. (54)

The second term in Eq. (54) corresponds to the energy losses in the elements and the coupling length, l_{i j}, denotes the length of the junction between the considered elements. The equivalent absorption lengths a_i and a_jof elements i and j relates the vibration reduction index to the structural reverberation time T_sof the elements and T_sdescribes decay of vibration energy. The relation between the equivalent absorption length and the structural reverberation time for element i is given by

a_i=2.2π²S_i cT_s,i

s fre f

f , (55)

where f_{re f} is a reference frequency taken as f_{re f} = 1000 Hz [10]. Combining Eq. (54) with Eq. (52) the expression used in the standard EN 12354-1 [16] for the flanking sound reduction index takes the form

Ri j=R_i+ R_j

2 + K_{i j}+ 10 log₁₀ SD

pS_jS_i

√a_ia_j

l_{i j} , (56)

which relies on available data for direct transmission of the individual elements, R_iand R_j, data for the vibration reduction index, data for the corresponding equivalent absorptions lengths a_iand a_j, related to the structural reverberation time T_s through Eq. (55), as well as geometrical data for the constructional arrangement.

Having data for a specific constructional arrangement allows us to use the EN 12354-1 model to predict the attenuation of sound propagating through various paths in the building structure, that is, we can obtain R_{i j}. The sound incident on the structural elements in the source room can for instance be represented by an audio signal and we want to listen to the transmitted sound inside the receiving room. Thus, we want to use the predicted sound transmission for manipulating audio signals to include the attenuation caused when propagating throughout the building structure. To use the predicted sound transmission R_{i j}for attenuating audio signals we then need some knowledge about signals and signal processing, which will be the topic of Section 2.4.

2.4 Signal theory for auralization

A signal can be considered as a time dependent function of a scalar physical quantity, for instance, sound pressure [10]. Let s(t) and s(n) denote a continuous time signal and its discrete representation, respectively, where the discrete time signal s(n) is obtained by sampling of the continuous time signal at discrete times.

The sampling rate, f_s, then relates sample n to the corresponding time value of the signal and the discretiz- ation allows for representing signals digitally serving as a requisite for computer based signal processing.

In the following when introducing central concepts of signal processing we will for simplicity stick to the continuous domain.

2.4.1 Fourier transform

Apart from the time domain, a signal may be expressed in the frequency domain. The frequency domain representation, or, the spectrum of the signal is denoted S( f ) and relates to the time domain representation as

S( f ) =F {s(t)} = ˆ _∞

−∞

s(t)e^{−2πi f t}dt, (57a)

s(t) =F⁻¹{S( f )} = ˆ _∞

−∞

S( f )e^{2πi f t}d f , (57b)

whereF {·} and F⁻¹{·} are the Fourier transform and the inverse Fourier transform respectively.

2.4.2 Convolution, impulse response and transfer function

Apart from the Fourier transform, a fundamental operation in signal processing and thus for auralization, is convolution. The convolution procedure concerns linear time-invariant (LTI) systems and provides the response of a system for an arbitrary input. An LTI system is linear in the sense that it obeys superposition, that is, a linear combination of input signals fed to the system results in that the output is the same linear combination of corresponding output signals. Time-invariant can simply be understood as time shifting the input signal by a certain amount results in that the output signal is shifted in time by the same amount. The time-variant property results in that the reaction of the system is independent of the time when excited. The relation between LTI systems and acoustics is that most systems transmitting sound can be approximated by LTI systems [10] and in acoustics, a transmitting system regards for example air or structures in which sound can propagate.

The response of an LTI system is described in the time domain by its impulse response, denoted h(t), such that given an input signal s(t) the output signal g(t) is given by

g(t) = s(t) ∗ h(t) = ˆ _∞

−∞

s(τ)h(t − τ)dτ, (58)

with ∗ being the convolution operator. The convolution operation may as well be carried out through multiplication in the frequency domain

G( f ) = S( f )H( f ), (59)

where the response of the system is given by its transfer function H( f ) =F {h(t)} and where the corres- ponding output, g(t), is given by g(t) =F⁻¹{G( f )}.

2.4.3 Retrieving an impulse response – Dirac delta function

With integrals of the types as in Eq. (57) and (58), the Dirac delta function δ (t) plays an important role.

The Dirac delta function is a generalized function satisfying the property ˆ _∞

−∞

δ (t)dt = 1, (60)

and may be interpreted as

δ (t) =

(+∞, t= 0

0, t6= 0. (61)

From the property of the Dirac delta function in Eq. (60) the impulse response or transfer function of a LTI system may be retrieved from excitation with a Dirac pulse, because it follows that convolution with a Dirac pulse results in the output

δ (t) ∗ h(t) = ˆ _∞

−∞

δ (τ)h(t − τ)dτ = h(t), (62)

and the Dirac delta function has a constant unit spectrum F {δ(t)} =

ˆ _∞

−∞

δ (t)e^{−2πi f t}dt = 1, ∀ f , (63)

such that the corresponding operation in the frequency domain

F {δ(t)}H( f ) = 1 · H( f ) = H( f ), (64)

yields the transfer function of the system. The concept of the Dirac pulse becomes useful for signal pro- cessing and auralization in, for example, room acoustics. Consider the case where the transmitting system is the path from a sound source to a receiver inside a room. The input and output signals represents the sound source and the sound at the receiver, respectively. Begin considering an empty room with completely

absorbent walls, the sound reaching the receiver is composed of the direct sound attenuated by the sound transmitting medium, a single pulse. Now let the boundaries be reflective, such that the sound reaching the receiver is composed of various reflections in addition to the direct sound, delayed due to the longer path covered as well as reduced in energy by absorption from the surfaces reflected off of in addition to the attenuation by the transmitting media. By letting the sound source, or input signal, correspond to a Dirac pulse the impulse response can be retrieved, by for instance measurement, for the specific configura- tion: room situation, source and receiver position. The retrieved impulse response is called a room impulse response(RIR) and the retrieved impulse response then allows for obtaining the output for the specific con- figuration for an arbitrary input signal through convolution, a basis for auralization considering its purpose of producing audible sound fields from numerical data. The information contained in the RIR allows for determining room acoustical parameters such as the reverberation time and deriving the reverberation time from a RIR is a topic that we cover in Section 2.5 whereas the discussion regarding certain characteristics of a RIR is continued in Section 2.6.

2.5 Reverberation time derivation

Having retrieved a room impulse response, the reverberation time can be obtained from the corresponding decay curve. The decay curve is defined in various ISO standards for reverberation time determination, for instance in ISO 354 and ISO 3382-2 [18, 19]. It is suitable that we here go through the definition of the reverberation time. Say that you have a sound source exciting a room such that an arbitrary steady state sound pressure level is reached, from the instant that the sound source is switched off, the time for the steady state sound pressure level to decrease by 60 dB defines the reverberation time. The decay curve should simply be understood as the energy of the sound field at time t after switch off. The most intuitive way for obtaining a decay curve is through measurement via the interrupted noise method, recording the level decrease from steady state after switch off [18]. However, the focus here concerns reverberation time retrieval from a RIR using the integrated impulse response method where the decay curve, denoted by E(t), is obtained from reversed time integration of the energetic impulse response, h²(t).

The procedure for obtaining E(t) can be divided into two steps, first, the total energy of the steady state sound field is calculated, a constant E_ss, obtained from integration of the energetic impulse response as

E_ss= ˆ _∞

0

h²(τ)dτ. (65)

The decay curve at time t, E(t), is then obtained by subtraction with a term corresponding to the energy components from the source being switched off to t seconds after switch off as

E(t) = E_ss− ˆ _t

0

h²(τ)dτ = ˆ _∞

0

h²(τ)dτ − ˆ _t

0

h²(τ)dτ

= ˆ _∞

t

h²(τ)dτ = ˆ _t

∞

h²(τ)d(−τ).

(66)

The result is the energy remaining t seconds after switch off [10] and the form of the right-most expression in Eq. (66) properly illustrates the name: reversed time integration. The reverberation time is given by the slope of the decay curve when expressed in decibel scale

E_dB(t) = 10 log₁₀ E(t) E_ss



. (67)

The slope, d_dB, is obtained by fitting a line to E_dB(t) using linear regression, for example, a least squares fit as proposed in the standard ISO 3382-2 [19]. The usual reverberation time is with reference to a 60 dB decay, because such a large decrease often can be infeasible to come across in measurement, several corresponding reverberation time measurements are used instead. The alternative reverberation time meas- urements are T₁₀, T₂₀and T₃₀, where the two latter are the most common and specified in ISO standards [18, 19]. The different reverberation time measurements are based on linear extrapolation of the decay rate, estimated for a shorter range and extrapolated to 60 dB, such that the quantities are analogous to the ordin- ary reverberation time T = T₆₀. We consider T₂₀as an example, the decay rate d_dBin dB/s is determined