11119 computational modelling of acoustics space

8/2/2019 11119 Computational Modelling of Acoustics Space

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Svensson and Kristiansen Computational Modelling

AES 22 nd International Conference on Virtual, Synthetic and Entertainment Audio 1

COMPUTATIONAL MODELLING AND SIMULATION OF ACOUSTICSPACES

U. PETER SVENSSON AND ULF R. KRISTIANSEN

Acoustics Group, Department of Telecommunications, Norwegian University of Science and Technology, NO-7491Trondheim, Norway

[email protected] , [email protected]

The computational modelling of acoustic spaces is fundamental to many applications in auralization/virtual acoustics.The demands vary widely, from real-time simulation in multimedia and computer games, to non-real time situationswith high accuracy needs, such as prediction of room acoustic conditions in music performance spaces. Acoustic spacesinclude single room or multi-room spaces, with simple or complex geometries and boundary conditions. Outdoor spacescan range from city environments to open landscapes. Sound transmission through partitions is an important issue insome cases. This presentation gives an overview of techniques used in the various auralization applications and with anemphasis on indoor spaces. Aspects of accuracy and computational efficiency will be discussed, as well as whichacoustical phenomena can and cannot be modelled with various techniques.

INTRODUCTION

Computational modelling of acoustic spaces has beengaining a wide interest and found more and more appli-cations. The first applications seem to have been themodelling of concert halls for room acoustic predictionand the possibility to auralize, or listen to, the resultshas always been central, [1-5], and the term "Auraliza-tion" emerged from this field. Interestingly, this earlyapplication is still among the most demanding onessince the purpose is to reproduce and evaluate quitesubtle aspects of the sound field so the accuracy is veryimportant. The rapid development of multimedia tech-nology has lead to the inclusion of acoustic modelling ina more general context, for instance in the standardi-zation work of MPEG [6],[7]. Different applicationsrange from high to low demands as regards accuracy, aswell as interactivity, as exemplified by the following.• Room acoustic prediction has high demands on

accuracy but low demands on interactivity.• Electroacoustic prediction. Modelling of large and

complex loudspeaker systems with high demandson correct modelling of source directivity and inter-action with the acoustic space. Interactivity de-mands are usually low.

• Video games on personal computers and special-ized consoles include acoustic modelling to variousdegrees and are good examples where plausible,rather than accurate, sound fields are the aim. Inter-activity demands are high.

• Music processing. Room acoustic modelling has

been used for the creation of naturally sounding re-verberation in commercial reverberation units.Plausible rather than accurate sound field modellingis usually desired. Interactivity demands might behigh.

• Virtual reality, where the quality of the acousticsimulation might be important for the sense of presence, or immersion. Accuracy demands mightbe high in, e.g., critical training situations where re-alistic acoustical cues are important. Other applica-tions might have low accuracy demands. Interacti-vity demands are often high.

The term virtual acoustics is often used for the acousticsubset of general virtual reality techniques, [6], and itincludes modelling of • The source,• The acoustic space,• The receiver.

In this paper, different techniques that are available for the modelling of acoustic spaces will be reviewed. Themodelling of the source and the receiver will be dis-cussed briefly.A general approach that is usually taken in acousticmodelling is based on the impulse response (IR) of thespace, with the computation/prediction of the impulseresponse being separated from the convolution/soundreproduction stage, see Fig. 1. Furthermore, differentreproduction techniques are available; indicated as

binaurally over headphones, binaurally over loud-speakers using crosstalk cancellation, and finally





Convolution with

2-N IRsSourcesignal

(monauralor multi-channel)

Reproduction

matrix

BinauralBinauralwith CTC

Multi-channel

ConvolutionIR computation

*

Figure 1: Auralization system employing source-chan-nel-receiver modelling of an acoustic space, and with

different reproduction options.CTC = crosstalk cancellation.

multi-channel loudspeaker reproduction. For dynamicauralization, the impulse response must be updated inreal-time which implies both a recomputation of theimpulse responses, and an interpolation in the convolu-tion process.Throughout this paper, the IR will be used in quite ageneral sense. It should be noted that, as indicated inFig. 1, the IR must be at least two-channel, for binauralreproduction techniques, or multi-channel for other reproduction techniques, which is discussed further in

Section 4.It can be observed that the two-fold calculation effortsare reflected in, for example, software for room acoustic

prediction. Since the impulse response contains all theinformation about the acoustic space (albeit in an ob-scured form), the analysis of the impulse response issufficient to some degree. Commercially available soft-ware for room acoustic prediction consequently oftenoffer the convolution stage, or auralization stage, as op-tional. In the following sections, different techniques for the IR computation in Fig. 1 will be discussed.

1 ACOUSTIC MODELLING

All kinds of acoustic spaces might be encountered inacoustic modelling tasks, such as single rooms thatmight be large or small, multiple rooms that are con-nected via various openings, outdoor situations in builtenvironments or in open landscapes. For auralization

purposes, a very wide bandwidth is required for a natu-ral sound, and in practice it is difficult to apply a singlemethod to all cases, and to the entire frequency range.This is especially true when there are high demands onfast processing. In the next section, some general as-

pects of modelling will be discussed first, such as per-ceptual and computational weights of the time and

frequency dimensions. In the following sections differ-ent techniques will be divided into two different classesof methods; those based on sound field decomposition(direct sound, wall reflections etc) and those based onsolving the total sound field rigorously. Only time-do-main calculations will be discussed here, even if fre-

quency-domain formulations might be used for convenience.For all modelling of linear systems, one could find theIR through a direct time-domain calculation or via aninverse Fourier transformation of a frequency response.An important point is that a frequency domain solutiontypically cannot make a distinction between early andlate parts of an IR. Since the late part usually is morecomplex, computation-wise, but less important percep-tually, see Fig. 2, this could lead to severely sub-optimalcomputations. In Fig. 2, the perceptual weight is illus-trated as being greatest for the direct sound and veryearly reflections, whereas the details of the later parts

become less and less important. Depending on whatcalculation method is used, the computational weight isat best constant, as indicated by the dashed straight linein Fig. 2 (for time-iterative calculation schemes, such asfinite-differences in the time domain) or increasing ac-cording to some power law, the dashed curve in Fig. 2(e.g., ray/beam tracing; or the image source method). Itis clear that there is a clear mismatch between computa-tional load and perceptual importance , for a straightfor-ward algorithm implementation.

Time

Weight per time unit

Perceptual

Computational

Figure 2: Qualitative representation of the perceptualand computational weights (for best/worst calculationschemes) as a function of time of an impulse response.

The reason for the increasing computational load asfunction of the impulse response time is the well-knownfact that in a closed room, the number of wall reflec-tions that reaches a receiver per time-unit is roughly

proportional to time squared. This leads to a computa-tionally higher and higher load as we try to determinethe later part of an IR, when we use a method that is

based on sound field decomposition, such as the imagesource method. At the same time, we should acknowl-edge that the sound field becomes so complex that wecannot perceive the details anymore and we perceive adiffuse sound field. Computational methods must





exploit this fact in order to be efficient, otherwise mostof the computation power will be spent on perceptuallyless important parts of the sound field. In Fig. 3, asimilar representation of the computational loads asfunction of the frequency range studied is presented.The diagram also shows a qualitative curve that indi-

cates our hearing's decreasing frequency resolution for higher frequencies. Methods based on sound field de-composition typically depend quite little on the studied

bandwidth (the dashed straight line in Fig. 3) whereasmethods based on a rigorous solution lead to dramati-cally increasing calculation times when the bandwidth isincreased (the dashed curve in Fig. 3).

Frequency

Frequency resolution of hearing Computational

Figure 3: Qualitative representation of the computa-tional weights (for best/worst calculation schemes) as afunction of frequency, as well as the frequency resolu-

tion of our hearing.

Physical model-based methods might aim at solving thewave equation with specified boundary conditions, thatis, to find the complete solution to the sound field. Thisis done by dividing the space into volume elements andthe wave equation is solved by some numerical solutionmethod (such as finite elements or finite differences).Alternatively, the volume problem can be transformedinto a surface problem. This is done by dividing the

boundary surface surrounding the space into surfaceelements and using the numerical boundary elementmethod for finding the solution to the sound field.Instead of solving the wave equation, known solutionsto the wave equation can be exploited by decomposingthe total sound field into elementary waves. Both exactsolutions and asymptotically correct (i.e., for high fre-quencies) might be possible to use. The image sourcemethod is a typical example of this.Methods based on wave field decomposition offer sig-nificant advantages. One of the biggest advantages isthat the elementary waves will appear as if they arrivefrom a distinct direction towards the receiver, and alsoas if they were emitted from the original source in adistinct direction. This makes it easier to separate thetreatment of the source, and the sound presentationtechnique, from the IR computation task. Furthermore,

being discrete-entities by nature makes it possible togive the elementary waves different weights and priori-

ties which is very attractive for real-time applications.The disadvantage is that the wave field decompositionusually is an approximation. Notably, however, we canusually only apply a complete wave field solution for deterministic situations/geometries. As argued later, thequasi-stochastic process that is introduced by, for exam-

ple, surface scattering, is such an important phenome-non that the ideal, purely deterministic situation isirrelevant. Rather, the situation is a mix of deterministicand quasi-stochastic which must be handled.Methods for wave field decomposition also suggestmuch simplified approaches of perceptually basedmodelling [8]. If the purpose is to recreate a perceptionof being in an acoustic space, whether it should be aspecific one or a more generic one, there are shortcutscompared to a detailed physically based modelling. Adetailed modelling of the direct sound and a number of distinct early reflections, with the right amount of dif-fusitivity and a balanced reverberation field might fulfil

many of the perceptual factors such as localization of the source, perception of room size and spaciousness,timbral/spectral qualities and temporal definition. At thesame time, all the effort that has been put into the de-velopment of reverberation units for music processingindicates that processing artefacts can be verydisturbing. Blesser has presented a very thoroughcoverage of the music processing application [9].

2 METHODS BASED ON SOUND FIELDDECOMPOSITION

We start by looking at methods based on sound fielddecomposition. Fig. 4 illustrates how an acoustic spacecan be represented by different types of "equivalentsources" - image sources (IS), that represent specular reflections, edge sources (ES), representing edge diff-raction, and surface sources (SS) that represent diffusereflection. These equivalent sources conceptually radi-ate in free space and are, in principle, supposed to fulfilthe boundary condition of the acoustic space. It should

be noted that this kind of sound field decomposition,while being very advantageous for auralization pur-

poses, might, or might not, be correct from an analytical point of view. The most common approach is the image

* **

*IS

IS

*ES

*ES

***

SS

Figure 4: Sound field decomposition: a real acousticspace is represented by different types of equivalent

sources. IS = image source, ES = edge source, SS = sur-face source.





source method which is based on geometrical acousticsand thus is generally correct only as a high frequencyasymptotic solution. However, the image source solu-tion gives exactly the correct solution for a rectangular room with rigid walls [10]. Furthermore, for a non-rec-tangular room with flat rigid walls, the image sources,

complemented by edge sources, give a correct solution[11]. Finally, the numerical solution of the Kirchhoff-Helmholtz Integral Equation using the boundary ele-ment method (BEM) employs boundary elements whichcould also be viewed as equivalent sources of thesurface source (SS) type in Fig. 4.The various equivalent sources will give a list of ele-mentary waves that arrive at the receiver. For each entryin the list there will be the following data:• Source data (directivity data)• Propagation data: frequency dependent attenuation

due toDistance spreading

Accumulated absorption loss at wall reflectionsAccumulated scattering loss at wall reflectionsAir absorption

• Receiver data (reception angle)For a sequence of specular reflections, this could beexpressed in terms of the arriving sound intensity I

I P eDF

r S

mr S j j

j

nrefl= −( ) −( )∏−

=41 12

1π α δ (1)

where P S is the sound source power, r is the total dis-tance, m is the air attenuation coefficient, DF S is the di-rectivity factor of the source, α j is the absorption factor of reflection j (in the sequence of n refl specular ref-lections) and δ j is the diffusitivity factor of reflection j.Since many of the factors in eq. (1) are frequency-de-

pendent either a single IR could be modelled for each of the elementary waves that have been identified, in order to give the same frequency dependence, or a set of pre-computed IRs (wall filters, source directivity filters, air attenuation filters etc, [6]) could be convolved.For specular reflections, this approach is very straight-forward since a specular reflection is represented by animage source, marked by IS in Fig. 4. For diffuse re-flections, marked by surface sources, SS, in Fig. 4, asimilar expression as in eq. (1) will result, except thatthe diffusitivity factor δ , rather than (1- δ ) should beused. Furthermore, depending on how diffuse reflec-tions are handled, more scale factors might be needed.Finally, edge sources that represent edge diffractionwaves, ES in Fig. 4, are less straightforward to model,

partly because they are diffracted from along edges. Inorder to have well-defined source radiation and receptor angles, the edge diffraction must be assigned to a singlesource along an edge, or a number of edge sources must

be placed along the edge.

The three different equivalent source types introducedin Fig. 4 and discussed above represent the three im-

portant phenomena: specular reflection, diffuse reflec-tion and edge diffraction. They will be discussed inmore detail in the following sections. One importantobservation at this stage is that in an acoustic space

there will be all possible mixes of these three phenom-ena, so the elementary waves must handle such mixed behaviour.

2.1 The image source concept

The concept of image sources is applied to various field problems in electromagnetic and acoustic wave propa-gation. It is based on the principle that a source in thevicinity of boundaries can be represented by the sourceand so-called image sources radiating in free space, asindicated in Fig. 5. The amplitudes of the image sourcesare adjusted so that the boundary conditions of the wallsare fulfilled at the positions in free space that representthe walls. If two semi-infinite planes are connected, andthe interior corner that is constructed has an angle of 180, 90, 60, 45, 36, ... degrees, such an image sourcerepresentation can fulfil the boundary condition exactlyif the walls are either perfectly rigid (the Neumann

boundary condition of zero velocity) or ideally soft (theDirichlet boundary condition of zero pressure). For allother corner angles, an edge wave is needed in additionto fulfil the boundary condition, as discussed further insection 2.5.These cases have primarily academic interest since werarely find such idealized cases in reality. The imagesource concept has been extended to arbitrary geo-metries with plane walls, and a number of papers haveapplied this to acoustics [10], [12-16]. Fig. 5 illustratesthe image source method for a source near two reflect-ing surfaces that are connected at a corner. The imagesources IS 1 and IS 2 are constructed by mirroring thesource in the two planes. A second-order reflection will

be represented by the image source IS 12 which is con-structed by mirroring the image source IS 1 in plane 2. Areceiver will experience an image source if the image

*** 1

2

IS*IS

1IS2

12

IS visible

IS , IS visible

IS , IS ,ISvisible

1

1 2

1 2 12

Figure 5: The image source method illustrated for asource and two reflecting planes.





source can be seen through the reflecting plane, that isto say, if the reflection point is inside the finite plane.For higher order reflections, all consecutive reflection

points must be checked for this. Furthermore, it mustalso be checked if there are any obstructions (other

planes) along the reflection paths.

In Fig. 5, the different zones indicate which of the threeimage sources that a receiver would experience. Thisillustrates two important aspects of the image sourcemethod (ISM):• First, the image source visibility must be checked,

along with obstruction tests, as mentioned above.The computational burden of these tests grows ex-ponentially with reflection order since the potentialnumber of image sources grows by a factor givenby the number of planes for each higher order,which is a well-known problem with the imagesource method. Such a computational growth is il-lustrated in Fig. 2. A number of methods have been

developed to overcome this straight-forward ap-proach. Many of the potential image sources can betossed out, exemplified by the image source IS 21 notbeing drawn in Fig. 5. IS 21 would represent a re-flection first in plane 2 and then in plane 1. This isan impossible combination since IS 2 can not seeplane 1 through its reflection plane, plane 2. As aconsequence, all the image source "children" thatcould have been generated by IS 21 will never comeinto play. This pruning of the image source tree de-creases the possible number significantly. So-calledhybrid methods use ray-tracing [17] or beam-trac-ing [18], [19] to find the possible image sources

and this is efficient even if some of the techniquesmight miss some specular reflections. It should benoticed, however, that high-order specular reflec-tions are of very limited interest since in realisticsituations, surface scattering and edge diffractioncomponents will overshadow the specular reflec-tions after just a few reflection orders [20], [21].

• A second important aspect of the image sourcemethod illustrated in Fig. 5 is that the sound fieldwill be discontinuous - every time the receiverpasses a zone boundary, one reflection will sud-denly disappear or emerge. This is clearly non-physical and one of the signs that the image sourcemethod must be an approximation, even if it isasymptotically correct for high frequencies. In sec-tion 1.2.5, on the edge diffraction concept, it isshown that waves that apparently emanate from theedges correct this discontinuity. Interestingly, suchedge diffraction waves disappear for interior cor-ners of 90, 60, 45 degrees etc.

It could be noticed that the image source method easilyis extended to directional sources (or receivers). For thespecial cases with interior corners that do not generateany edge diffraction, the image source solution is exact

even for directional sources. One interesting aspect of the ISM is that a receiver can be mirrored instead of thesource. This might be advantageous, for instance if areceiver is stationary and the source is moving. Fur-thermore, it has been shown that so-called bi-directionaltracing of the possible specular reflection paths is more

efficient computation-wise [19].A last important issue for the image source method iswhen boundary conditions are more realistic than ide-ally rigid or ideally soft. Fig. 6 illustrates an imagesource above a single reflecting plane, with a given lo-cally reacting impedance.

*

*

Q

Q IS

• θ 0

Figure 6: A source above a locally reacting plane.

The image source could be given the amplitude and di-rectivity given by the plane wave reflection frequency-domain expression

Q Q R R IS ω ω ω ω ζ ω

θ ζ ω

θ

( ) = ( ) ( ) ( )=( )−

( )+0

1

1, cos

cos

(2)

where Q0 and Q IS are the source strengths of the originalsource and the image source, respectively, and R is the

plane wave reflection factor. The angle θ is as indicatedin Fig. 6 and ζ is the locally reacting specific impedanceof the plane surface. For the modelling of enclosedacoustic spaces where a diffuse sound field builds up,there are so many reflections that the detailed directivitymight be completely irrelevant. Then an average reflec-tion factor is often derived from the diffuse fieldabsorption coefficient α ,

R ω α ω ( ) = − ( )1 (3)

It can be noted that only the magnitude of R can be de-rived this way. A phase function can be constructed

based on the assumption that the reflection is a mini-mum-phase system. Reflections are not generally mini-mum-phase since, e.g., any delayed reflection or wave

propagation in the plane of the reflecting surface wouldviolate the minimum-phase assumption. However, typi-cal flat surfaces with low absorption tend to yield mini-mum-phase reflections. Available data for absorptioncoefficients are usually given as octave band values,from 125 Hz to 4 kHz, of the diffuse field α , see[22]. For use in the IR computation stage, one needs toderive discrete-time "wall filters" that represent the wallreflection factors by modelling digital filters that fit the





desired frequency dependence. As described in [6] thiscan be done by precomputing all possible cascades of wall combinations for the first few reflection orders.As pointed out above, eq. (2) is an approximation and itfails when the source or receiver is close to the surface(relative to the wavelength) and/or the angle θ is close

to 90 degrees. This is obviously a problem if we want tomodel long-distance propagation above ground sincemany situations use sources and receivers close to theground. The realism of a simulation of an outdoor situa-tion might depend quite strongly on the accurate model-ling of the comb filter effect caused by interference

between direct sound and ground reflection.Many papers on outdoor sound propagation have stud-ied this ground reflection, [23]. The general solutionrelies on a complicated integral formulation whichmight be too computationally demanding for many uses.A precomputation can be carried out and a moreaccurate image source could be used, that is to say, a

more accurate reflection factor R in eq. (2) could be de-rived. As shown in [23], if ζ happens to be real-valued,a more accurate image source strength can be derived

but it has the impractical property of being dependentnot only on the angle θ but also on the distance to thereceiver. No simplified formulations can offer high ac-curacy for the special cases of long-distance propaga-tion above the ground. This more complex behaviour,compared to the simple image source concept, actuallyimplies that a surface impedance leads to surface scat-tering, since the reflected wave will not appear as if it isemanating from a single image source.

2.2 Diffuse reflection / ray-tracingOne big drawback of the image source concept is that itcannot handle diffuse reflection/surface scattering. Dif-fuse reflection is caused by surface roughness and/or surface impedance and it can be illustrated as in Fig. 7.A part of the reflected sound wave makes up a specular reflection, as discussed in the previous section, but therest of the reflected sound wave is reflected in a diffusefashion. The exact nature (e.g., the directivity) of thisdiffusively reflected sound depends on the geometricaland acoustical details of the surface.

*

*

Q

Q IS

0

Figure 7: Illustration of surface scattering. An incidentsound wave is reflected in the form of one specular

component, represented by an image source, andscattered components.

The ray tracing technique is illustrated in Fig. 8. A largenumber of rays are emitted from the source, and as theyhit a surface they are either specularly or diffusely re-flected. A receiver is defined as a volume, typically asphere, and if a ray hits the receiver, one reflection isadded to a list. The principle is very simple, and it has

been widely used in room acoustics, [24-28], as well asin underwater acoustics and outdoor sound propagation[29]. In the latter applications refraction caused by windand temperature gradients, as well as turbulent scatter-ing and extended reaction impedance planes, are han-dled by more sophisticated algorithms. One special case

*Q 0

Receiver

Figure 8: Ray-tracing - reflection at a surface, viaspecular reflection (solid line) or diffuse reflection

(dashed line)

is worth noting: for a linear vertical sound speedgradient, the rays will travel along circular paths.The principal behavior indicated in Fig. 7 would suggestthat each ray should be split up at each reflection, sothat one ray, carrying a fraction of the reflected sound

power, is reflected specularly whereas a number of raysare reflected according to some scattering distribution.The incident power is then divided as

P P P Pin abs spec refl diff refl= + +. .

= + −( ) −( )+ −( )[ ]Pin α α δ α δ 1 1 1 (4)

where α is the absorption coefficient and δ is the diffusereflection coefficient. Such a computation schemeclearly grows rapidly and it is not desirable. Dalenbäck has suggested that such a scheme could still be handled

by grouping hits on walls into discrete wall elementsand by storing each hit for later group re-emission [30].This way all combinations of specular and diffusereflection could be handled in an iterative manner.Other approaches have been reviewed in [31-32].The most common approach is, however, to let eachincident ray be reflected either specularly or diffusivelyand this is used in most ray-tracing algorithms in roomacoustics. This can easily be controlled by the genera-tion of a random number between 0 and 1, and if thisrandom number is smaller than the value δ in eq. (4),then the reflected ray is treated as diffuse and its direc-tion will be chosen stochastically according to somedesired distribution function. The most common func-





tion has been Lambert's law, [22], which states that thereflected intensity I refl is distributed as

I refl refl∝cos θ (5)

where θ refl is the reradiation angle as in Fig. 6. The scalefactor that is needed will depend on whether a singleray, or multiple rays, are reradiated.Much work has been done on surface scattering andusually, stochastic surface variations are assumed. Mostanalysis methods are based on the so-called Kirchhoff approximation which assumes that the sound pressure atthe surface of the reflecting surface is twice the pressureof the incident wave. This is a good approximationwhen slowly undulating surfaces are studied. A com-mon assumption for this approximation to hold is thatevery point of the sea-bottom can be seen from thesource and the receiver, which puts a limit on how steepgradients the surface can have. Embrechts, [33], hasapplied these techniques to the room acoustics case. Thestrength of the surface scattering depends on the ampli-tude of the surface roughness compared to thewavelength. As illustrated in [33], the scattering coeffi-cient increases rapidly with frequency. This leads to a

problem for some ray-tracing algorithms: if a stochasticchoice between specular and diffuse reflection is used,then it might be necessary to trace each octave bandseparately.There has been a great interest lately in surface scatter-ing in room acoustics and standardized methods are de-veloped by ISO for determining the so-called scatteringcoefficient s, [34], and by AES for the diffusion coeffi-cient d , [35]. The scattering coefficient is a measure of how much the reflected energy that is non-specular whereas the diffusion coefficient offers detailed data onthe directivity of the scattering. In room acoustics sur-face scattering might be caused by statistical roughness

but also by deterministic geometrical features along thewalls. Such non-stochastic scattering can not be mod-elled by Lambert's law, and this has lead to thedevelopment of the more detailed measurements in-volved in determining the diffusion coefficient. As of today very few measured values of scattering propertiesare available for acoustical modelling. The standardizedmethods are the first step in the direction of makingsuch data available.The importance of modelling diffuse reflection properlywas identified in the first larger evaluation of computer-

based RIR prediction [36], since rooms with similar absorption properties but different scattering propertieswill have significantly different reverberation times[22], [37-39]. Calculation methods that do not modelscattering must then apply artificially high absorptionfactors to wall surfaces to get realistic reverberationtimes. Furthermore, all-specular sound fields produce avery synthetic and unnaturally sounding simulation of adiffuse sound field.

2.3 Radiosity methods

A technique which is very closely related to ray tracingis the radiosity method. Instead of launching a largenumber of rays and letting them sample the boundarysurfaces, radiosity methods subdivide the boundariesinto smaller elements. Rays are then sent between these

predefined surface elements. The contribution strengths,so-called form factors, are calculated for each element-to-element combination, as illustrated in Fig. 9. Anintegral equation can describe the intensities incident oneach element as a sum of the contributions from allother elements, and an original source. This method has

been used for a long time in studies of heat radiationand lighting, and also in computer graphics. A number of papers have described the application of this method,and developments, to acoustics, [18], [40-43].

Element i

Element j r i j θ i

θ j

Figure 9: Illustration of the radiosity method

The basic integral equation, for a discretized model, can

be described as

I t I t k I t r

cSi i j i j j

j i

j j i j( ) = ( )+ −

∑→ →

→≠

0 ρ :

∆ (6)

where the form factor k, for completely diffusely re-flecting walls according to Lambert's law, is

k r

j ii j

ij→ =

cos cosθ θ

π 2 (7)

and I i(t ) is the incident sound intensity at element i, ρ isthe energy reflection coefficient, and I t i0→ ( ) is the

contribution from the direct source. The sum in eq. (6)can be solved iteratively. It is possible to extend this

basic formulation in order to include partly specularlyand partly diffusely reflecting surfaces, [44] by derivingmore complicated form factors. Here it should be no-ticed that the elements do not need to be smaller thanthe wavelength, they merely need to be small enoughthat the form factor doesn't vary too much across theelements. Some important points for both ray tracingand radiosity are:• The basic formulations, such as Lambert diffusion

indicated by eq. (7), assume that the reradiation





angle is independent of the incidence angle. Morecomplicated relationships are possible, and neces-sary in order to include specular reflection as wellas diffuse reflection.

• The intensity at the surface is assumed to be thesame as in the incident beam. This is the Kirchhoff

diffraction approximation and it is true only for anelement which is part of a large surface. Edge dif-fraction effects lead to that the surroundings of eachelement modify the field at the surface, as discussedin the section on edge diffraction. More compli-cated form factors could in principle take such aneffect into account but the form factors would befrequency dependent.

2.4 Beam tracing

As mentioned earlier, the beam tracing method is anefficient method for finding the valid image sources in aacoustic model. Beam tracing has evolved from raytracing, as illustrated in Fig. 10. A number of rays areemitted from the source, and adjacent rays are treated asconstructing a beam, a cut-out of the wave-front, as in-dicated by the incident beam in Fig. 10. Each ray must

be checked for plane-hits and a hit with one of the finitereflection planes causes a specular reflection of the ray.When two adjacent rays hit different walls, new"children" rays are generated so that the incident beamis split up into two reradiated beams, as in Fig. 10.Whenever a receiver is inside one of the beams, a hit isregistered, and such a hit represents a valid imagesource, i.e. a specular reflection-combination. Early ver-sions of beam tracing were approximate in the sensethat they used single rays, as in ray tracing, with at-tached circular or triangular cross-sections that weregrowing along the propagation [45], [18]. No split-up of

beams when they hit an edge was done. Such a split-uphas later been explored in [46], [47] and [19].The beam-tracing algorithm is more efficient than theclassical image source method for finding the allowedimage sources, but even larger computational gains can

be done by subdividing the volume into sub-domains.The idea is that rays must be checked for hits with

planes, as in ray tracing, and it is very redundant tocheck all rays against all planes. A subdivision leads toan efficient search tree algorithm as described in [19].It could be noted that such a sub-division might be moreefficient the less open a geometry is, that is to say, a

Incident beam

Split-up andreflected beam

Split-up beam

Thinreflecting plane

Figure 10: The beam-tracing method

corridor-like, or city street geometry, rather than a con-cert-hall geometry.

2.5 Specular reflection + edge diffraction

As mentioned in previous sections, the image sourceconcept leads to unphysical discontinuities in the sound

field whenever a receiver is moved across a zone boundary as was indicated in Fig. 4. Also for the beamtracing method, it can be observed in Fig. 10 that thesplit-up beams yield discontinuous wavefronts. This isalso exemplified in Fig. 11(a) where a stage housegeometry, from [48], is illustrated.

(a)

*

(b) 40 50 60 70 Time [ms]

(c) 40 50 60 70 Time [ms]

Figure 11: (a) A stage house model with a source and areceiver array indicated. Stacked impulse responses for

(b) specular reflection only, and (c) specular reflection +edge diffraction.

An array of receiver positions is positioned laterally infront of the stage house opening and the IRs are calcu-lated for these positions. When these IRs are plotted in astacked fashion, the wavefronts can be identified, as

presented in [49]. The truncated wavefronts that result





from the geometrical acoustics-based image sources areclearly visible in Fig. 11(b).Interestingly, it is possible to formulate the sound fieldas a sum of these truncated image source contributions,and so-called edge source contributions. Fictive sourcesalong all edges will radiate a complementary edge dif-

fraction sound field which has a rather complicated di-rectivity and that adds to the image source field so thatthe total sound field is exactly correct and therefore, thetotal sound field is perfectly continuous when zone

boundaries are crossed. This is clearly shown in Fig.11(c) where the truncated wavefronts in Fig. 11(b) have

been corrected.Two important aspects can be identified, first the obvi-ous effect that edge diffraction yields a sound wave thattravels around corners. A second effect is that the edgediffraction spreads in all directions and modifies thesound field significantly also in the illuminated region.The edge sources can be implemented by subdividing an

edge into edge elements and letting all of these consti-tute a secondary source with a directivity function β asgiven by, [11],

β νϕ

ν η νϕ = ( )

( )− ( )∑ −=

sin

cosh cosh cos

i

ii1

1

4(8)

where

ϕ π θ θ i S R= ± ± , η α γ α γ

= +1 sin sin

cos cos(9)

The factor ν is the so-called wedge index, which is de-fined as ν = π /θ w, where θ w is the wedge angle (on the

open side). Whenever the wedge index is an integer value, the directivity function is zero. As an example,for an interior 90-degree corner, θ w = π /2, so ν = 2 andthus, there is no edge diffraction. Furthermore, in eq.(9), the term ϕ i is created by the four possible combina-tions of plus and minus signs. Finally, the angles θ S , θ R,α , and γ represent the angles for a sound path incidenton, and reradiating from, an edge point, as detailed in[11]. This directivity function has been derived from theexact Biot-Tolstoy solution [50] that was exploited byMedwin and others [51-52],[48]. Some points can bemade about the formulation in eqs. (8) and (9),• The directivity function is somewhat complicated

but as shown in [11], more efficient formulationsare possible.

• In addition to the directivity functions the edgesources cause a delay and a spherical spreading at-tenuation, as given by the distance to, and from, thepoints along the edge.

• A numerically efficient implementation can dividethe edge into equally sized edge elements, smallerthan half the shortest wavelength. For highest accu-racy, the directivity function would need to be inte-grated for each such element, but in most cases a

very simple integration holds because of the gentlevariation of the directivity function. An exception,however, is when the receiver comes close to azone boundary, i.e., when the specular reflectionsuddenly disappears or emerges, as in Fig. 11 (b).Then a more careful integration is needed.

• In the same fashion as for image sources the visi-bility of these edge sources must be checked, as il-lustrated in Fig. 12 for a stagehouse geometry. In[21], it is discussed how such visibility test can bedone more efficiently in the context of a beam-tracing algorithm.

• The wedge index ν is implicitly included in the ex-pression in eq. (8). This has the important conse-quence that the edge diffraction wave will alwaysdepend on the two planes that make up a wedge.This means that the total reflection off a plane(specular + edge diffraction) can not be viewedalone; the plane's surrounding planes will affect thereflection.

•

•IS

S

RVisibleedgefrom IS

Invisibleedgefrom IS

Figure 12: Visibility of edge sources.

The approach to combine specular reflections and edgediffraction has been used earlier in acoustics and elec-tromagnetic wave propagation. Earlier simplified for-mulations were the Geometric Theory of Diffraction(GTD), [53], and the Uniform Theory of Diffraction(UTD), [54]. However, both are high-frequency asymp-totic solutions where UTD has a precisely predictablelow-frequency limit [55]. Other frequency domainformulations that are high-frequency asymptoticallycorrect, have been suggested in [56], [57].The clue to a formulation that works for the entire fre-quency range, as in [11], is a decomposition into secon-dary edge sources along the entire edge of a plane, thatis to say, a line integral formulation, rather than just aradiation from a single so-called apex point of an edge.However, such a decomposition is also used in methods

based on the Kirchhoff approximation, which states thatthe sound pressure at a surface is exactly twice the inci-dent free-field pressure. Many calculation methods,such as in underwater acoustics, are based on this ap-

proximation. A very elegant line integral formulationresults, as presented in [58], but as shown in [59], the





Kirchhoff approximation has the unfortunate propertythat it does not give a solution which is correct at highfrequencies. Rather, it gives accurate results for diffrac-tion directions that are close to perpendicular to the

plane, but gives larger and larger errors for increasingangles.

One negative side effect of the fact that the entire edgesreradiate is that there is no distinct incidence angletowards the listener. If efficiency is crucial, thediffracted wave may approximately be assigned to asingle point along the edge, [48], [60].The edge diffraction concept suffers from the same ex-

ponential computation time growth that was identifiedfor the image sources. In [20] and [21], it was demon-strated that the number of edge diffraction contributionsactually surpasses the number of specular reflectionsafter some reflection orders. This is a sign that a high-order specular reflection is not very useful in itself sinceit will always be accompanied by edge diffraction

waves.Similar to the previous discussion on combinations of specular and diffuse combinations, there will be all pos-sible combinations of diffraction and specular reflectionand these combination components will dominate moreand more for higher and higher reflection orders. Thediffracted sound field components can be viewed as

*

*

S

IS

IR

Time

3 4 5 6 7 [ms]TF Freq.

0.2 2 20 [kHz]0 dB

-7 dB

Figure 13: Illustration of the edge diffraction's polarityswitching effect. (a) An example of a source-receivegeometry near a corner. (b) The IR and the transfer

function corresponding to the specular reflection plusthe edge diffraction wave, normalized to the specular

reflection amplitude.

more diffuse than the specular components because theedge sources radiate in all directions. Thus the edge dif-fraction transform energy from specular into (quasi-)dif-fuse energy. While the surface scattering effect isstronger and stronger the higher the frequency is, theedge diffraction effect leads to more and more diffusi-

tivity the lower the frequency. It has been suggested thatthis edge diffraction could be handled by assigning anartificially high surface scattering coefficient at low fre-quencies, or specifically when a reflection hit point isclose to a plane edge. However, the edge diffraction hasa fundamentally different functioning because it is re-flected as a wave of opposite polarity relative to thespecular reflection. This is illustrated in Fig. 13, and thisopposite polarity leads to a high-pass filtering effect thatcannot be implemented by artificially high surfacescattering coefficients. One consequence of the edgediffraction's polarity switching effect is that the sound

pressure at a reflecting surface becomes weaker close to

the edge. This is one of the inaccuracies involved informulations used in radiosity and ray tracing where it isassumed that the sound pressure at a surface is twice theincident sound pressure.A common problem in acoustical modelling is how tomodel curved surfaces in geometrical-acoustics basedmodels. Contrary to what one intuitively would think,results will not be more accurate if a curved surface ismodelled with finer and finer plane surfaces. The reasonis that the edge diffraction components will be increas-ingly important as reflecting surfaces get smaller.

2.6 Transmission through walls

Sound transmits through walls via vibrations that might be transversal waves (fulfilling a wave equation, thusyielding non-dispersive waves) if the partition is mem-

brane-like. For more solid partitions, which is the com-mon case, more complex bending waves will begenerated. The bending waves are excited by incidentwaves and reradiate waves in a reciprocal fashion, both

back into the sending space and into the receiver space.Bending waves are dispersive which means that theycan not be modelled by ordinary image sources. A firstattempt could, however, be to use transmission coeffi-cients for a wall partition and model a wall transmissionIR to fit the data, and apply it to an ordinary imagesource solution. The directivity of a transmitted bendingwave is very special and this might be important tomodel more carefully for receivers close to walls.

3 GENERAL SOLUTION OF THE WAVEEQUATION

The previous section dealt with sound field decomposi-tion that employed explicit solutions to the wave equa-tion. The more general problem is to solve the waveequation with its specified boundary conditions (wallgeometry and properties etc). This can be done by using





analytic solutions, which is available only for a very fewsimple cases, or by numerical solution methods. Thelatter can either divide the space into volume elementsor the surrounding boundary surface into surface ele-ments. As indicated in Fig. 3, the computation effortsincrease rapidly with increasing bandwidth and these

methods are therefore primarily used for the low-fre-quency range.There have been a number of attempts to apply this typeof general solution to the low-frequency range, and touse other approximate methods for the mid-to-high fre-quency range [61-63]. This makes sense not only fromthe computational perspective but also because the de-tailed sound field that a general solution method offersis wasted at high frequencies, at least as long as thesound field contains several interfering components.The limited frequency resolution of our hearing makes itimpossible for us to notice the detailed interference

patterns at high frequencies.

For outdoor sound propagation, analytic solutions mightoffer useful solutions. One common example is whenthere is a linear temperature gradient above ground, asmentioned in the section on ray tracing. An increasingtemperature gradient leads to the well-known effect thatsound can travel over surprisingly long distances. Con-versely, a decreasing temperature gradient leads to thatthe wavefront is bent upwards thus not reaching posi-tions on the ground.Another example where analytic solutions might be use-ful is for small rooms since they tend to be rectangular in shape. A rectangular room is one of the few caseswhere an analytic solution is available, in the form of a

modal sum which in the time domain can be formulatedas in [64].

3.1 Volume element methods

A number of numerical solution methods exist for solving the wave equation by subdividing the air vol-ume into volume elements. The elements must be smallenough, with at least N elements per wavelength stud-ied, where N depends on the used method and it cantypically be 6-10, or even higher. Typically, more re-fined formulations might work for larger elements. Sev-eral of the methods can handle arbitrary partialdifferential equations, so including, for example, me-dium losses, non-linearities and other phenomena isstraightforward.As pointed out in the introduction, the focus here is ontime-domain formulations. Such formulations are basi-cally time-iterative: they calculate the correct soundfield at each time instant from the known one at the pre-vious time instant, where the time step is the samplingfrequency used.The volume-element based methods have in commonthat if the bandwidth (given by the sampling frequency)is doubled, then the element size must be halved which

basically leads to that eight times as many elements areneeded in a three-dimensional model. Furthermore,twice as many time steps must be calculated. Ideally thecalculation time would then increase by the fourth

power of the bandwidth increase factor and the memoryrequirements would increase by the third power. These

estimates are true for the simple algorithms that use atime-iterative scheme where the number of operations per element stays the same regardless of sampling fre-quency and number of elements. In addition, somemethods might require extensive precalculations, for example to set-up matrices like in the finite elementmethod. An overview of several of the methods men-tioned here is given in [65].These methods are typically only used at low frequen-cies and combined with other methods for mid- to highfrequencies. This is both because of the rapidly in-creasing computation time and memory requirements,and because of the decreasing need for detailed infor-

mation on the sound field for higher frequencies.Sources are typically represented by prescribing someinitial sound pressure distribution around a source posi-tion according to its directivity. At low frequencies,some sound sources can be modelled as very smallcompared to the wavelength and then the sound pressurecan be specified in free space. Larger sources mightneed to be modelled as reflecting objects with specifiedvelocity distributions.A specific problem for volume element methods is themodelling of open spaces. Much effort has been put intodeveloping absorbing boundary conditions so that anopen-ended air volume can be truncated and modelled

with such a boundary conditions.For a receiver in the form of a human head, low fre-quency modelling could simply capture the sound pres-sure at two points in free space at a distancecorresponding to the distance between the two ears.More accurate models of the human head would bequite much more complicated to implement.

3.1.1 The finite difference method in the time domain(FDTD) and the digital waveguide method

Finite difference formulations in the time domain leadto very simple algorithms for solving the wave equation,

based on approximating differential operators with dif-ferences. In one dimension the double gradient of thesound pressure p, for a position x = xi, at sampled timeinstant n, could be approximated by values of p atneighbouring positions, a discrete step ∆ x away, but atthe same time instant n,

∂ ∂

2

21 1

22 p n

x

p n p n p n

xi i i i( ) ≈ ( )− ( )+ ( )

( )+ −

∆(10)

A three-dimensional formulation is straightforward toderive as well. Furthermore, the double time derivativethat is needed for the wave equation would be,





∂ ∂

2

2 2 21 1 2 1 p n

t c

p n p n p n

t i i i i( ) ≈ +( )− ( )+ −( )

( )∆(11)

By combining eqs. (10) and (11), and by dividing thethree-dimensional volume into cubical elements, thesound pressure values at the N elements for time instant

n+1 can be calculated from the values at previous timeinstants via a relationship such as

p n p n p ni i+( ){ }= ( ){ }− −( ){ }1 1A (12)

where { p} is a transposed vector of N elements and A isa matrix of size [ N , N ] which has non-zeros in a few po-sitions per row. For a simple finite-difference formula-tion such as in eq. (12), or the basic digital wave-guideformulation in [66], there would be 6 or 7 non-zero val-ues per row of the matrix A . These numbers correspondto that the value of p at one element is affected by the 6nearest neighbour elements and possibly the elementitself, from the previous time instant, see Fig. 14. For astable iteration process, the sampling frequency must behigher than a certain critical value. An obvious problemwith cubical elements is that boundaries must be formed

by stair-case approximations. Non-cubical elementswould be possible to use, but more complicated algo-rithms than the one indicated by eq. (12) would be re-quired. Rigid-wall boundary conditions areimplemented by stating that two adjacent cell layershave identical values of p (i.e., the pressure-gradient iszero). More complicated boundary conditions require aconvolution formulation for the boundary layers, relatedto the impedance impulse response. Of special interestare absorbing boundary conditions, [67], which makes it

possible to model semi-infinite domains like outdoor spaces. In [67] and [68], a staggered, or interlaced,mesh, was used. This means that the sound pressure iscalculated at the center of each element whereas the

particle velocity is calculated at the surfaces of the ele-ments, in the three perpendicular directions. The sim-

plest FDTD formulations typically require 10-20elements per wavelength for high accuracy.A few observations can be made about FDTD formu-lations.• There is a well-known dispersion error which

means that the propagation speed will be different

in different directions [66],[67]. This can be im-proved substantially by using an interpolated meshwhich lets the value of p at one element be affectedby more of the neighbor elements. In [66], the 26closest elements were used which lead to that thematrix A in eq. (14) would have 26 non-zero ele-ments. Other possibilities are to use higher-orderapproximations of the differential operators thanthose in eqs. (12) and (13), [69], which also leads tomore non-zero elements in A, or to use triangularelements rather than cubical [70].

• Another observation is that the dispersion will be

frequency dependent, and that the dispersion errorwill increase with frequency. In [66], a resamplingand frequency warping of the input and output sig-nals was used and lead to small dispersion errors.

• A third observation is the very important featurethat the matrix A contains a few constant values per

row, and that these values don't need any precal-culations. This leads to a very efficient algorithmand in [62] a large concert hall was modelled with0.25 m large elements and a sampling frequency of 2.4 kHz. In total 0.6 million elements were used forthe hall model.

Figure 14: The element arrangement for a finite differ-ence model using cubical elements. A central element

and its 6 closest neighbor elements are indicated.

The computational load for an FDTD formulation likethe one above is straightforward to estimate since analgorithm such as in eq. (12) leads to a constant number of operations per element, regardless of sampling fre-

quency and number of elements. Consequently, the cal-culation time increases by the fourth power of the bandwidth increment factor.

3.1.2 The finite element (FE) method

An FE-model divides the modelled air volume intosmall elements that can take non-uniform shapes. Thesound field inside each volume is described in the formof an analytical shape function, such as linearly varyingor a quadratic function. By forcing the sound pressure to

be continuous at element boundaries, and by forcing thesound field inside the element to fulfil the wave equa-tion in some averaged sense, it is possible to write thesound pressure at the N elements as [65],

102

2

2c

p

t pi

iK M∂ ∂

+ { }= (13)

where K and M are two [ N , N ] matrices that are derivedfrom the geometrical shapes of the elements, the shapefunctions used and the material properties. It should benoted that the matrices K and M have very few non-zero elements since they describe how neighbouringelements are affecting each other. The time-dependencecan be solved using a finite-difference scheme. For the





simplest finite difference technique such as in the sec-tion on the FDTD technique, a matrix equation like

p n p n p ni i i+( ){ }= ( ){ }+ −( ){ }1 1B (14)

yields the sound pressure values in a time-iterative way.The matrix B is derived from the K and M matrices, as

B E K M= − −2 2 2 1c t ∆ (15)

where E is a unity matrix. It is noteworthy that eq. (14)has the same form as eq. (12) for the FDTD technique.However, the precalculation of the K and M , and thenthe B matrix adds significantly to the calculation timecompared to the simpler FDTD method. The inversionof the K matrix becomes a matrix with more values thanthe K matrix itself so the B matrix will contain manymore values than the matrices for FDTD methods. Thesimilarity between these methods reflects the fact thatthe finite difference method can be seen as a specialcase of the finite element method [71].Another common approach for deriving the time-do-main response with the FE method is to use a modaltechnique. For lightly damped systems, the mode func-tions in the entire system can be derived first, together with the modes' eigenvalues. Then the damping con-stants for the modes can be derived in a second stage,[72]. Such a method is, however, of less interest for IR computation when computation time is critical.A general observation for the FE method is that theelements can be larger than for the simplest FDTD for-mulations, due to the inherent wave field distributioninside the elements. Also, the non-regular element

shapes can fit the boundaries without any staircase approximations. It is not clear, however, how these factorsaffect the IR accuracy. In the same way as for the FDTDmethod, the need to model semi-infinite spaces has leadto the development of semi-infinite elements. An alter-native to semi-infinite elements is to describe the soundfield inside the volume in terms of, for example, spheri-cal harmonics, [73], which can be matched to perfectlynon-reflecting boundary conditions. This leads to a sig-nificantly more complicated algorithm.The computation time for a time-iterative FE formula-tion such as in eq. (14), will grow with the bandwidthincrement factor to the power of four, but in addition

comes the set-up and calculation of the involved matri-ces which will add significantly to the time consump-tion.

3.1.3 The transmission line method (TLM)

The so-called transmission line method uses a cubicallysubdivided model of the acoustic space, just like theFDTD formulations described earlier. The time-iterativeformulation used is very simple, [65], [74] and [75], asillustrated in Fig. 15. Sound pulses are followed along

branches between nodes, and a pulse incident on onenode is scattered in all six directions. The sound pres-

sure is collected at each node by summing all the inci-dent sound pulse from all six directions.

1 13

13

1

3

13 1

3

23-

x x y

z z

Figure 15: The transmission line method.

Interestingly, as shown by Kagawa, [75], such a metho-dology leads to a formulation which is equivalent to theFDTD formulation in eq. (12). Consequently, the com-

putational effort will be the same as for FDTD. Bounda-ry conditions will be implemented somewhat differently

for the transmission-line method and the method is alsoconsidered as being more stable than FDTD methods.

3.2 Surface element methods

The Kirchhoff-Helmholtz integral equation (KHIE)gives the sound field in an interior or exterior space interms of contributions from the sound pressure andnormal particle velocity (or from a fictive layer poten-tial) on the boundary surface of the modelled space. Thenumerical solution of this KHIE is the Boundary Ele-ment Method (BEM). In the same way as for the vol-ume element methods, the surface must be subdividedinto elements that typically need to be smaller than ap-

proximately 1/8 of the wavelength. The BE method hasthe important advantage over volume element methodsthat open space need no special considerations. Time-domain formulations have been presented in [76-79]. Aslightly different formulation, as in [80], can be given interms of equivalent sources, where the boundaryelements are viewed as sources with a source signal(layer potential) q i(n), n being the time sample, and thenthe sound pressure at a receiver position R is given by,

p n p n g k q n k R R p i Rk

n

i

N i( ) = ( )+ ( )∑∑ −( )→ →

==0

01, (17)

where g k p i R, → ( ) is the time-domain Green's function,or IR, for the sound pressure p from element i to thereceiver position R, p n R0 → ( ) is the sound pressure atthe receiver as caused by the original source, and N isthe number of boundary elements. The situation is illus-trated in Fig. 16. The similarity between eq. (7), for theradiosity method, and eq. (17) could be noted. However,the difference is that the intensity, averaged over ele-ments that are large compared with the wavelength, isthe calculated quantity in radiosity. In BEM, the surfaceelements must be smaller than the wavelength, and the





sound pressure and particle velocity are calculated. For rectangular elements, as indicated in Fig. 16, the IR g pwould be the impulse response of a piston, see [80]. Thecomputationally heavy part for the BE method is to cal-culate the source signals q i(n) which must be done in afirst step. The receiver position is placed at the elements

and the signals q i(n) are adjusted so that the sound pressure and particle velocity at each element fulfil thespecified boundary conditions. The boundary elementscan be of monopole-type, dipole-type, or a combination.For the simplest case of rigid walls, and monopole-typeelements, the source signal q(n) can be the velocity of the piston, which then must be the same as the particlevelocity vn(n) in front of the piston. When the receiver isat the elements, the particle velocity is given by

v n v n g k q n k n j n j v i jk

n

i

N i, , ,( ) = ( )+ ( )∑∑ −( )→ →

==0

01(18)

where g k v i j, → ( ) is the IR which gives the particle ve-

locity at element j, caused by element i emitting a pulse,and v nn j,0 → ( ) is the contribution from the original

source to element j. Eq. (18) yields an integral equation,

v n v n v nn n n( ){ }= ( ){ }+ ( ) ( ){ },0 0G

+ ( ) −( ){ }+ ( ) −( ){ }+G G1 1 2 2v n v nn n ... (19)

where{ v(n)} is a transposed vector of N elements, con-taining all the elements' source signals at time instant n,G (0) is an [ N , N ] matrix of the impulse response valuesfor time instant 0, G (1) a similar matrix containing all

the IR values for time instant 1 etc. The matricesG

(k )state what each element contributes to all other elementswithin 0,1,etc time instants after a pulse is emitted fromthe element. A very important point is that most of theseIRs have an initial time delay, i.e., a number of initialzeros, caused by the propagation time from element toelement, and also that the IRs have just a few non-zeroelements. The length of a specific IR depends on thedifference in length between the shortest and longest

paths between two points on the two elements. Thenumber of IR matrices G (k ) will be given by the longestdistance between two points of the entire model. Intotal, there will be many IR matrices G (k ), but with few

non-zero values in each. Eq. (19) can give the solutionof { vn(n)},

v n v nn n( ){ }= − ( )[ ] ( ){ }+(−E G 01

0,

+ ( ) −( ){ }+ ( ) −( ){ }+ )G G1 1 2 2v n v nn n ... (20)

where E is a unity matrix of size [ N , N ]. The matrix thatmust be inverted has very few non-zero values outsidethe main diagonal since all elements "reach themselves"at time sample zero, but the influence from one elementto another within time sample zero is small, [81].

The computational load can be estimated by observingthat for a doubling of the bandwidth, the number of

boundary elements would typically increase by a factor of four, and the sampling frequency would be doubled.Then the length of each element-to-element IR would

basically double and the number of time steps to com-

pute would double as well. The calculations involved ineq. (20) would in total then increase at least by 16 times.In addition comes the time for the inversion of the ma-trix, and the calculation of all the matrix values.

Element i

Element j

Figure 16: Illustration of a boundary element model.

4 IMPLEMENTATION OF AN AURALIZA-TION SYSTEM

In sections 1-3, the computation of an impulse responsewas discussed in quite general terms. The aim was theimplementation into an auralization system, as presen-ted schematically in Fig. 1. Then the reproduction tech-nique, the source modelling and the potential need toupdate the IRs dynamically must be studied more indetail, which is done in the following subsections.

4.1 Reproduction techniques

One distinct class of reproduction techniques includesthose that are based on the binaural technology, [82],[83], aiming at reproduction of the two signals at thetwo ears of one or more listeners, using headphone re-

production or cross-talk cancelled loudspeaker repro-duction, [84]-[85].For the calculation methods in section 3 the binauraltechnique would yield two complete IRs, representingthe sound pressure at the two ears. For such a case, the"reproduction matrix" in Fig. 1 simply has to conveythose two signals to the two ears of a listener. For theheadphone reproduction, headphone equalization filters,e.g., would be needed in this matrix. This approach hasa distinct drawback: for dynamic auralization, if a lis-tener rotates his or her head, the complete IRs wouldhave to be modified dynamically. This can be done veryefficiently for some IR calculation methods but notalways. Since the IR calculation requires heavycomputational efforts it would be desirable to minimize





the blocks that need to be updated dynamically. Thecalculation methods in section 2, on the other hand,could yield a list of IRs that each represent a singleelementary wave, arriving from a distinct direction. Thismakes it possible to let the reproduction matrix takecare of the synthesis of the sound pressure at the ears of

the listener. Head-related transfer functions (HRTFs) for different incidence angles are stored in a library, and alistener's head-rotation can then be handled bydynamical updating only at this last stage and let the IR of the acoustic space remain constant. These possibleapproaches are illustrated further in Fig. 17, where theIR convolution block (from Fig. 1) is detailed. Fivedifferent stages are indicated for the source, room/space,and receiver modelling. Some of the outer stages can be"peeled" off and treated in separate blocks, e.g., thelistener rotation for the sound field decomposingtechniques.The second class of reproduction techniques aims at

reproducing the sound field over a little area or volume,in which listeners could move freely. These includeAmbisonics, [86], and Vector-Base Amplitude Panning(VBAP), [87], whereas the Wave Field Synthesis, [49],

Source rotationSource position

Receiver rotationReceiver position

Room/space data

Fig. 17: The IR convolution block, with the involvedsource, room/space, and receiver modelling stages.

is developed for reproducing a sound field over a wider area. Common for all of these is that the "receiver rotation" stage in Fig. 17 is not needed, and if the de-sired receiver movements are within the reproductionvolume, the "receiver position" layer isn't needed either.A reproduction matrix could be illustrated as in Fig. 18,with three different input signal formats and the threedifferent output signal formats indicated. Some combi-nations, e.g., a list of elementary wave inputs to themulti-channel loudspeaker output could representseveral different choices such as Ambisonics or VBAP.Ideally, the IR computation should be independent of the chosen reproduction technique but this is possibleonly to a certain extent. IR calculation methods that are

based on a sound field decomposition into elementarywaves would is the most flexible format since all de-sired reproduction formats could be synthesized. How-ever, if a complete IR has been calculated, it cangenerally not be decomposed into elementary waves.The stages in Fig. 17 that deal with the source are dis-cussed further in section 4.2.

Of all the different techniques that have been discussedin sections 1-3, not a single one has the potential for be-ing accurate and efficient across the entire frequency

Binauralsignal

•••

M signalsfrom distinctdirections

•••

Ambisonicformat,order L

To headphones

To two loud-speakers(using CTC)

•••To N loud-speakers

Reproduction matrix

Fig. 18: A reproduction matrix with three different inputformats and three different output formats.

range, and IR length, especially not when perceptualissues are taken into account, such as indicated by Figs.

2 and 3. Methods based on geometrical acoustics, withextensions, are very efficient for and most reasonablyserve as a basis for a real-time calculation procedure.Since the largest inaccuracies with these methods typi-cally occur for low frequencies, and the more accuratemethods suffer from a rapidly increasing computationtime as the bandwidth is increased, a band-splitting ap-

proach is quite an obvious choice, if the simplifiedmethods prove to be inadequate.As shown by [6] and [21], methods based on imagesources, and preferrably also implementing edge dif-fraction, can be made very efficient, also for real-timeapplications. Precalculation of geometrical data and

storing wall and source directivity filters etc can be employed to a large extent. Furthermore, priority-drivensearch for valid sound paths is interesting and couldyield a gradual degradation when computation resourcesare low.A challenge still seems to be to incorporate specular reflection, edge diffraction and surface diffusion in asingle efficient algorithm. As has been discussed,diffuse reflections are important both for a realistic esti-mate of the reverberant field and for a realistic percep-tion of the simulated sound field.More simplified, perceptually based, methods can offer direct sound plus a few reflections from a detailed

model but will often have problems with estimating rea-sonable parameters values for, e.g., the reverberant part.As regards the methods for solving the total field, sim-

ple and straightforward algorithms in the FDTD class,such as the interpolated algorithms presented in [65]seem very promising because of the lack of heavy pre-calculations in contrast to, e.g., time-iterative BEM or FEM formulations. By developing automated meshingalgorithms, geometrical models used in geometricalacoustics methods might be used directly in volume- or surface-element methods.





4.2 Sound source modelling

The modelling of the sound source is in principle sepa-rate from the acoustical space that is modelled but aseparation is not possible since the directivity of thesound source affects the IR. Very often a point source isused but primarily for convenience. All natural sound

sources, speaking humans, music instruments etc, havequite complicated source directivities [88]. This can behandled by modelling digital filters that fit measureddata [6], or to describe the directivity in sphericalharmonics functions [89]. Many such harmonics termsare, however, needed in order to fit the measureddirectivities of musical instruments. In large rooms withdominating diffuse fields, it is most important to at leastinclude a filter which corresponds to the frequencyresponse difference between the average directivity (thatis to say, the total radiated power) and the directivity inthe direction that the anechoic recording was made.A special concern is that some musical instruments donot have a well-defined directivity since a certainfrequency component might be radiated differentlydepending on how it was excited. The same frequencycomponent might be radiated for different notes, whichin turn can be radiated by different parts of the instru-ment, e.g., different organ pipes.In Fig. 17, it is indicated that the source data possiblycould be separated. In the same way as for the receiver modelling, IR calculation methods that yield elementarywaves that represent distinct source radiation directionswould be the most flexible, with a source rotation being

possible without affecting the room/space layer of theIR calculation. The special concern mentioned abovewith the source directivity not being a well-definedquantity might be solved using multi-channel dryrecordings, as indicated in Fig. 1.

4.3 Interactive auralization

Interactive auralization systems, as of today, are basedon the sound field decomposition techniques, as in theDIVA system [6], the Spatializer [8] or the system in[19] and [21]. For indoor spaces, the reverberation isoften done in the form of a much simplified reverbera-tion unit, the parameters of which are controlled fromthe IR computation. An obvious problem is situations

with atypical, non-diffuse and/or non-exponential reverberation fields.As discussed earlier, the potential need to use a

bandsplitting approach, and a more accurate method for low frequencies would require that the low-frequency

part is treated separately. Unfortunately, the incidenceangles are not available when the total sound field iscalculated. However, up to around 500-700 Hz, alistener's head affects the sound field very little so by

picking up the sound pressure at two points displaced by17-20 cm, a reasonable representation of the sound atthe ears can be computed.

5 EVALUATION/ACCURACY

The success of a specific modelling method all dependson how well it performs in the context of its use. In real-time applications, a gradual and robust technique for adapting to available computing resources might have ahigher priority than the naturalness of the reverberation,

for instance. Still, as has often been pointed out, in vir-tual training situations, it is important with high enoughrealism/accuracy that no computational artefacts supplycues that are not there in the real situation.For the room acoustic modelling there have been twointernational round robins evaluating a number of com-mercially available prediction softwares as well as re-search lab softwares. A first stage, presented in 1995[36], dealt with the modelling of a medium-sized audi-torium and only with the 1 kHz octave band. A number of parameter values were measured and predicted andthe deviation was quantified in terms of established Justnoticeable limens. One conclusion was that softwaresthat did not use diffusion had to overestimate the ab-sorption factors in order to get good values of the rever-

beration time. Another conclusion was that some parameters, such as the strength index, i.e., normalizedtotal SPL, was quite accurately estimated by most soft-wares. A second stage of this Round robin studied alarger concert hall, and the frequency range from the125 Hz to the 4 kHz octave band was evaluated in 2000[90]. One conclusion of this second stage was that errorsgenerally increased in the 125 Hz octave band, and thatsome receiver positions were very problematic becausethere was no sightline from the source to the receiver and then the calculation of many parameters break down. The state-of-the-art of prediction software at thatstage was typically a hybrid method with accurate im-age sources for the first few reflection orders, and raytracing or approximate cone/beam tracing for the higher orders, thus facilitating the combination of specular anddiffuse reflection.A third stage of the Round robin is running at the mo-ment, and it studies a smaller room, with some compli-cated diffusing elements, and with an optionalevaluation of auralization possibilities.So far, quite few studies have evaluated completeauralization systems. It is a sensitive evaluation sincethe sound reproduction stage introduces many factors tocontrol. It is clear, however, that there is a strong needfor further evaluation efforts.

6 CONCLUSIONS

An overview has been given of the most common tech-niques for computing physically based impulse re-sponses of acoustic indoor and outdoor spaces, for usein auralization and other virtual acoustics applications,real-time or non-real time.One class of methods included those based on soundfield decomposition, which may be accurate for some





situations and approximate for other cases. The acousti-cal phenomena of specular and diffuse reflections, aswell as edge diffraction and impedance surface reflec-tions, were discussed, and it was argued that the differ-ent techniques: the image source method, edgediffraction, ray-tracing, beam-tracing, and radiosity

covered these phenomena to different degrees. Another class was the more rigorous solutions of the wave equa-tion that solve the total sound field and the time-itera-tive techniques used in FDTD and digital waveguideswere found to be attractive because of their verystraightforward algorithms. The time-iterative BEM or FEM formulations could potentially be attractive butthey involve more complicated algorithms.It was argued that the rapidly increasing computationtime with increasing bandwidth for the more rigorousmethods are in contrast with the decreasing need for sound field details at high frequencies. Consequently, a

band-splitting approach seems inevitable when higher

accuracy is needed or problematic cases are studied,such as outdoor sound propagation over long distances.Further work is needed for finding the optimum cross-over frequency range, as well as rigorous algorithmsthat are as efficient as possible for the frequency rangeof interest. For the sound field decomposing methods,the challenge is to combine modelling of specular re-flection, diffuse reflection and edge diffraction in anefficient way that overcomes the exploding computationneeds for later parts of the IR, while still managing tomodel non-classic reverberation fields.

7 ACKNOWLEDGEMENTS

The authors thank the reviewers for their valuablecomments.

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