ACOUSTIC RENDERING OF BUILDINGS. Rudolf Rabenstein, Oliver Schips and Alexander Stenger

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1 ACOUSTIC RENDERING OF BUILDINGS Rudolf Rabensten, Olver Schps and Alexander Stenger Unverstät Erlangen-Nürnberg, Lehrstuhl für Nachrchtentechnk Cauerstrasse 7, D Erlangen, Germany e-mal: ABSTRACT Whle vsual renderng of buldngs s the state of the art n today's desgn programs, acoustc or audtory renderng s stll n ts nfancy. Ths paper revews some promsng approaches to the computer smulaton of sound propagaton and percepton n buldngs. The range of methods spans from the numercal soluton of the wave equaton to advanced geometrc methods based on ray tracng and radosty algorthms. Furthermore concepts for modellng the human sound percepton are dscussed. Fnally some ssues of practcal mplementaton wll be addressed. 1. INTRODUCTION It s a common feature of modern computer aded desgn programs to reward the user wth a photorealstc renderng of the prospectve buldng. However despte all vsual perfecton, the audtory component s completely mssng. There s no mpresson of the room acoustcs, no assessment of the nose level n offces or workshops and no evaluaton of the effect of sound absorbng materals by computer smulaton. It has to be emphaszed, that all these topcs are well studed subjects and that computer smulatons of ndvdual acoustc effects are common practce. The problem wth acoustc (or audtory) renderng of buldngs s that a realstc smulaton of the acoustc propertes requres a real tme computaton of complex wavefelds. In order to provde a Compact Dsk sound qualty, t would be necessary to update a 3D sound pressure feld wth complex boundares every 23 mcroseconds. To provde such a feature as an addton to a vsual renderng capablty s more than even modern desktop computers can handle. Nevertheless, there have been successful attempts from the vrtual realty communty to assgn sounds to vsual objects and to supply real tme mplementatons of vrtual scenes [6]. Whle such audo-vsual smulatons may sound plausble to the vewer of an anmated move, they do not satsfy the requrements of engneerng practce. However t seems worthwhle to step ahead by combnng proven vsual renderng algorthms wth elements of scentfc smulaton. Ths paper revews some promsng approaches n ths drecton. It starts wth a short ntroducton nto the bascs of acoustcs and systems theory and presents n some detal the concept of the room mpulse response. The core of the paper s the treatment of four approaches to the smulaton of sound propagaton n rooms. Fnally modellng of human sound percepton by head related transfer functons s dscussed and some mplementaton ssues wll be addressed. 2. SOUND PROPAGATION 2.1. Basc Theory From a scentfc standpont t s always pleasng to establsh a computer model from frst prncples. In acoustcs (and many other felds) ths amounts to the numercal soluton of partal dfferental equatons. From a practcal standpont, ths s not attractve at all and as a consequence, smpler models are preferred. We present here two general modellng concepts: the wave equaton and an nput-output model from lnear systems theory. Both concepts are based on the assumpton of lnearty: The sound feld from two sources n parallel s a superposton of the ndvdual sound felds of each sngle source Wave Propagaton and Acoustcs Sound propagaton can be descrbed by velocty and pressure as functons from space and tme. It s governed by the wave equaton c 2 p 2 (1) where p(x;t)s the sound pressure, x s the vector of space coordnates, t s the tme, and c s the speed of sound. The dervaton of the wave equaton from frst prncples s gven e.g. n [17]. In order to calculate the sound feld emanatng from a source n a specfc room, we need an addtonal source term n (1) and boundary condtons, whch descrbe sound reflecton and absorpton at the walls. A numercal technque for the approxmate soluton of the wave equaton s presented n secton

2 Lnear Systems Theory A completely dfferent approach to the descrpton of sound propagaton (and other lnear and tme-nvarant systems) s the general nput-output relaton from lnear systems theory y(t) =h(t)v(t)= Z t,1 h(t, )v( )d : (2) v(t) s the source sgnal (e.g. sound pressure) at the locaton of the sound source (see fg. 1), y(t) s the resultng sgnal at the locaton of some recever (e.g. mcrophone). The sound propagaton from the source to the recever s descrbed by the functon h(t). It s called the mpulse response, because t s the response at the recever to a very short pulse at the source. v(t) y(t) Fgure 1: Source and recever poston wthn a buldng v(t) h(t) y(t) Fgure 2: Input-output descrpton of sound propagaton wthn a buldng For fxed source and recever locatons, all the spatal nformaton s contaned n the mpulse response h(t). Thus, the knowledge of the mpulse response s suffcent to calculate the response y(t) to a known source sgnal v(t). The sound propagaton n a room accordng to fg. 1 s thereby reduced to the black-boxmodel gven n fg. 2. The operaton (2) scalled aconvoluton and denoted by. A smpler expresson results n the frequency doman Y (!) =H(!)V(!) (3) where Y (!) =Ffy(t)g s the Fourer transform of y(t) and smlar for V (!). H(!) =Ffh(t)g s called the transfer functon. It s a complex valued functon of the real frequency parameter!. The meanng of the transfer functons s best descrbed by the response to a sne-wave of angular frequency! v(t) = sn(!t) : (4) The response of a lnear and tme-nvarant system s agan a sne-wave, however, n general wth dfferent ampltude and phase shft y(t) =jh(!)jsn(!t + (!)) (5) Ampltude jh(!)j and phase (!) are just the magntude and phase of the complex transfer functon H(!) H(!) =jh(!)jexp(j(!)) (6) p Here j =,1 s the magnary unt. The transfer functon H(!) s a functon of the frequency!, snce the response of a system to sne-waves wth dfferent frequences wll n general depend on!. Transfer functon and mpulse response are equvalent descrptons of the black-box-system n fg. 2. Nether convoluton nor Fourer transformaton of contnuous-tme sgnals can be mplemented drectly on a dgtal computer. Instead one works wth sampled versons of v(t) and y(t). Ths turns (2) nto a dscrete convoluton and the Fourer transformaton nto a dscrete Fourer transformaton, whch can be effcently computed wth fast algorthms (Fast Fourer transformaton, FFT). Dscrete convoluton and FFT are the buldng blocks for the dgtal processng of sound sgnals (and others). A more detaled descrpton of dgtal sgnal processng technques n acoustcs s found n [9, 17]. In ths paper, we wll keep the contnuoustme notaton (2,3) for smplcty. The black-box-descrpton (2) s smple and effectve because all the spatal nformaton s concealed n the mpulse response h(t). The only remanng problem s how to obtan h(t). There are three possbltes: The mpulse response can be calculated from the Green's functon of the wave equaton (1). However, ths s only feasble for very smple geometres and very smple boundary condtons. There are relable and quck methods to measure the mpulse response of a room. Examples are presented n secton 2.2. Of course, measurements are restrcted to exstng buldngs. The mpulse response can be estmated n the plannng stage from the room geometry and other data. Several methods are presented n sectons 2.3.2, 2.3.3, and Measurement of Room Impulse Reponses Methods for the measurement of room mpulse responses are based on the transfer functon descrpton (3) of a room accordng to fg. 1. Note, that the general form of (3) s also vald for more complex geometres than the one depcted n fg. 1. Snce the source sgnal v(t) s known and the response y(t) can be obtaned by measurement, we can compute dscrete tme approxmatons to V (!) and Y (!) by the FFT-technques dscussed above. The mpulse response follows by dvson and nverse Fourer transformaton (also performed by FFT) Y(!) h(t) =F,1 V(!) (7)

3 Practcal measurement procedures have to consder the selecton of approprate source sgnals, background nose durng the measurement and possble nonlneartes of the electro-acoustcal equpment. A method for ths purpose whch has also been successfully appled to the analyss of other weakly nonlnear systems has been descrbed n [7, 8, 14]. Its performance s shown here by some measurement examples. Fg. 3 dsplays the source functon v(t), whch has been used n these measurements. It s a broadband sgnal wth bounded ampltude and s called a chrp sgnal. The mpulse response of an 8 m 3 anechoc chamber s shown n fg. 4. The rapd decay s due to the hghly absorbng walls. The mpulse response of an offce room wth a typcal duraton of 0.1 s s shown n fg. 5. Note the dfferent tme scales n fgs. 4 and 5. Fg. 6 dsplays the mpulse response of two nterconnected corrdors wth a total of 250 m 3 and no furnture. One of the corrdors s connected to a starcase. The drect sound path was blocked, so that only reverberaton occurs. Note agan the tme scale. chrp sgnal v(t) Fgure 3: Chrp sgnal as source functon mpulse response h(t) tme t mpulse response h(t) tme n seconds Fgure 5: Impulse response of an offce room mpulse response h(t) tme n seconds Fgure 6: Impulse response of a system of nterconnected corrdors some detal. The frst method s based on the numercal evaluaton of the wave equaton (1) and s capable of a fathful smulaton of all acoustcal effects. A severe lmtng factor s the computng power to perform these computatons n real tme. The other three methods are based on geometrcal acoustcs,.e. they model sound propagaton by sound rays whch travel smlarly to lght rays n optcs. Dffracton effects are neglected by ths assumpton. However, sound ray propagaton can be treated wth the same prncple methodswhch are used wthgreat success n vsual renderng Numercal Smulaton tme n seconds Fgure 4: Impulse response of an anechoc chamber Once obtaned by measurements, we can use any of these mpulse responses to mmc the response of the correspondng rooms by computng the convoluton wth an arbtrary source sgnal v(t) accordng to (2). Although ths s easly done on a small desktop computer, t s not true renderng, because t requres the buldng to be already avalable for measurements Modellng of Sound Propagaton To perform acoustc renderng of vrtual buldngs, we have to fnd means to compute the room mpulse response from nothng more than the buldng geometry and the acoustc propertes of the buldng materals. Four methods to perform ths task wll be dscussed n Conceptually, the most straghtforward way to the smulaton of the acoustc behavour of a buldng s the numercal soluton of the wave equaton (1). We present a smple fnte dfference approach for a rough sketch of the dea. Detaled accounts on the numercal soluton of partal dfferental equatons can be found elsewhere [11]. The basc steps are the representaton of the sound pressure p(x;t)on a dscrete-tme, dscrete-space mesh wth a fnte number of nodes and the approxmaton of the dfferental operators n (1) by dfference operators defned n terms of the mesh nodes. For smplcty, we restrct the dervaton to one spatal dmenson,. e. p(x;t)=p(x; t) wth the scalar space varable x. The nodes of the mesh are gven by x = nh and t = kt, whereh and T are the step szes n space and tme, respectvely, and n and k are the correspondng dscrete varables. The node values of p(x; t) at these locatons are denoted by p n;k. A smple second order approxmaton of the dfferental

4 operators s obtaned T 2 pn;k+1, 2p n;k + p n;k,1 p 2 1 h 2 pn+1;k, 2p n;k + p n,1;k (8) (9) When the values of the sound pressure p n;k for the tme steps k and k, 1 are already calculated, then we obtan an explct expresson for the approxmate value p n;k+1 at the new tme step k +1by replacng both sdes of (1) wth ther dscrete tme approxmatons (8,9) p n;k+1 = 2 p n,1;k+2(1, 2 )p n;k + 2 p n+1;k,p n;k,1 ; (10) where = ct (11) h s the rato between the dstance ct that the sound wave travels wthn one tme step of length T and the spatal step sze h. The most smple formula results for =1 p n;k+1 =, p n,1;k + p n+1;k, pn;k,1 : (12) A graphcal representaton of ths dscretzaton scheme s shown n fg. 7. The value p n;k+1 s obtaned by the sum of the neghbours of p n;k and by subtractng p n;k,1. k+1 k k-1 n-1 n n+1 Fgure 7: Second order dscretzaton scheme for the wave equaton The generalzaton of ths dscretzaton scheme to three spatal dmensons s straghtforward. Wth the vector n =[n 1 ;n 2 ;n 3 ] of dscrete space varables n each drecton and wth 2 =1=3follows p n;k+1 = 1 3 m p m;k, p n;k,1 : (13) P Here m denotes summaton over all spatal neghbours of p n;k. Ths dscretzaton scheme has been used n [12, 13] for the smulaton of ar pressure wave propagaton n a model of three rooms separated by open doorways. It has been shown that the effects of reflecton at the walls and dffracton at a pllar could be numercally reproduced. A major drawback of ths method s the numercal expense assocated wth the update of the 3-D mesh n every tme step. To llustrate the stuaton, we estmate the computng power n floatng pont operatons per second (flops) whch s necessary for a real tme smulaton of the propagaton of a sound wave wth a frequency f 0 of 3 khz n a buldng wth the volume of 100 m 3. The maxmum tme step sze accordng to the samplng theorem s T =1=(2f 0 ). In practce, T should be smaller to avod numercal dsperson. Snce the update of one node value accordng to (13) requres 7 floatng pont operatons, the computng power for one node s gven by N 0 =14f 0 =42kloflops. Furthermore, the step szes n tme and space are lnked by the requrement 2 = 1=3, whch yelds the spatal step sze (c 340m=s) p p 3 c h = 3 ct = 10 cm 2 The number M of nodes n the volume V s M = V=h 3 = 100=(0:1) 3 =10 5, whch amounts to a total computng power of N = MN 0 =4:2Ggaflops for ths rather modest stuaton. Snce the number of flops grows lnear wth the volume V and proportonal to the fourth power of the frequency f 0, we cannot expect real tme renderng of buldngs by numercal smulaton n the near future. However, numercal smulaton s feasble for low frequences, and t s the only method whch descrbes dffracton correctly Image Sources We have just seen, that real tme smulaton by fnte dfference approxmaton of the wave equaton has a prce that s hard to pay. We can cut ths prce, f we are wllng to sacrfce some of the benefts of physcal modellng. In a descrpton by the wave equaton, any pont n the sound feld can act as a source and vce versa, we can pck up the sound at any pont of the wave feld. Such a flexblty s not always requred. A mnmal approach would be, to consder only one source and one lstener. Also some acoustcal effects can be neglected under certan crcumstances. For example, f the room szes are large compared to the wavelength, dffracton plays only a mnor role. In most practcal stuatons, ths s true for frequences above 1 khz. If also reflecton at the walls s consdered n a smplfed way, then a geometrc model of wave propagaton can be adopted. The great advantage of ths approach s, that geometrc models are well establshed n vsual renderng and consequently some proven algorthms have been developed. However, acoustcal renderng requres some addtonal consderaton, whch s not necessary n vsual renderng. The most mportant topc s the fnte propagaton speed of sound waves compared to the almost nstantaneous spread of lght. Furthermore, dampng of lght ntensty s neglgble for small dstances, whereas sound waves undergo severe attenuaton. The adapton of vsual renderng schemes for acoustcs has to take these effects nto account. The smplest way of geometrc modellng s the method of mage or mrror sources. It s a classcal f 0

5 method n physcs for the treatment of boundary value problems. An early mplementaton as a computer program for acoustc smulaton has been descrbed e.g. by [1]. Fg. 8 shows a rectangular room wth one source and one recever. The propagaton s modelled as the superposton of drect, sngle and multple reflected rays from the source to the recever. Rather than constructng all the dfferent propagaton paths wthn the room, the reflectons are modelled as drect paths from addtonal magnary sources outsde the room. They are postoned n such a way, that ther assocated drect paths concde wth the reflectons nsde the room. These postons can be obtaned by symmetry consderatons. The arrangement n fg. 8 models only a few of the frst reflectons. For a realstc smulaton, t would have to be extended to an nfnte contnuaton n three dmensons. R The response to an arbtrary source sgnal v(t) follows from (2) and (15) y(t) = 1 4d v(t, d c ) (16) as the sum of attenuated echos wth delay d =c. Angle and frequency ndependent wall absorpton can be modelled by ncluson of approprate absorpton coeffcents b ` whch descrbe the absorpton of the sgnal from source at the `-th reflecton. h(t) = b p (t); b = L Y `=1 b ` (17) L s the total number of reflectons for source. The model can also be extended to frequency dependent reflecton, when the reflecton coeffcents b ` are replaced by the mpulse responses b `(t) of systems whch descrbe the reflecton processes h(t) = b (t)p (t); (18) where S 1 S b (t) =b 1 (t)b 2 (t)b L (t) (19) S 2 S 3 S 4 Fgure 8: Modellng of wave propagaton by mage sources (recever R, source S, mage sources S, =1;2;3;4;:::) For a smple vsual model, we can magne the room n fg. 8 to be a mrror cabnet wth deally reflectng walls. The response to an mpulse at the source s then smply the sum of delayed mpulses, where the delay tme s gven by the total length of the reflecton path dvded by the speed of sound. The length of the reflecton path s obtaned from fg. 8 as the dstance of the correspondng mage source to the recever poston. For acoustc modellng, we have to consder the mrror sources as emtters of sphercal pressure waves. The sound pressure at the recever caused by the source wth number at dstance d emttng a pulse (t) s then gven by (see [1]) p (t) = 1, t, d 4d c (14) When all walls are deal reflectors then the total mpulse response n terms of sound pressure s gven by h(t) = p (t) (15) where the sum s taken over all surroundng mage sources wthn a certan regon (deally nfntelymany). s a multple convoluton model for the frequency dependent reflecton at a total of L surfaces. The descrpton of these multple reflectons s smpler n the frequency doman. We obtan by Fourer transformaton the transfer functon H(!) = B (!)P (!); B (!)= L Y B `(!) `=1 (20) wth the transfer functons B `(!) of each frequency dependent reflecton at the `-th surface. The mage source model s conceptually smple, but t requres a hgh number of mage sources n order to model the complete mpulse response of a room. If we want to compute the mpulse response of a room wth volume V up to a length of t 0 seconds, we have to extend the room n ether drectons untl the mage sources fll a sphere of radus ct 0. The number of mage sources s roughly gven by the relaton of the volume of the sphere to the volume of the room 4 N = 3 (ct 0) 3 (21) V For t 0 =1sand V = 100 m 3, a total of 1.7 mllon mage sources results. Ths number can be handled for frequency ndependent reflectons, but the consderaton of frequency dependent reflectons by multple convolutons becomes mpractcal. The numercal expense ncreases also when multple sources have to be consdered. Addtonally, any change of poston of

6 the recever or one of the sources requres a recalculaton of the mpulse response. Furthermore, obtanng the locatons of the mage sources s a tedous procedure f the buldng geometry s more complcated than the rectangular box model shown n fg. 8. Ths procedure s descrbed n [3] for polyhedra wth arbtrary shape. The locatons of possble mage sources are computed by a tree-structured algorthm and tested for valdty, proxmty and vsblty. Only vald sources whch are vsble and suffcently close to the lstener contrbute to the mpulse response. However, the computng tme requred to carry out ths procedure s a severe lmtng factor. On the other hand, the mage source method s capable of modellng the nterference between reflected sounds, f the model s formulated n terms of pressure rather than ntensty [4] Ray tracng The ray tracng method s also based on geometrcal acoustcs. It s smlar to the mage source method, snce t also consders the possble reflecton paths by whch the sound travels to the recever. The source model emts a number of sound rays n all drectons. A certan amount of energy s assgned to each ray. The propagaton path of each ray s traversed and the energy losses due to absorpton n the ar and at each reflecton are recorded. All rays that ht the recever wthn a certan amount of tme contrbute to the mpulse response. Ths procedure s very smlar to the well known applcaton of ray tracng n vsual renderng, except for the dfferent physcal effects, whch account for the energy losses along the path. The advantage of the ray tracng method over the mage source method les n the treatment of arbtrary complex 3D shapes wthout requrng excessve computng power. The numercal modellng of the reflectons s done smlar as n the mage source method. The effect of deal reflectons, frequency ndependent and frequency dependent absorpton can be calculated by the same prncpal algorthms as n (15,17,18) Radosty The mage source method and the ray tracng method depend on the poston of the recever. Whenever the recever changes the postonwthna buldng, the mpulse response has to be computed anew. The radosty method s a vew ndependent algorthm. It requres knowledge about the buldng geometry and about the sound sources. From ths nput, an approxmaton of the sound feld wthn a buldng can be constructed. The radosty method has been used for a long tme n thermodynamcs to calculate the radaton heat transfer between surfaces [16]. Later, t became one of the workng horses of computer graphcs for vsual renderng [5]. Its adapton to audtory renderng s descrbed n [15]. When appled to lght waves, the radosty method descrbes an equlbrum for the energy dstrbuton wthn an enclosure, that s constructed from a fnte set of surfaces (patches). Ths fnte set of patches covers the surface of the enclosure completely, so that all energy flows are consdered. Any openngs are also descrbed as a partal surface wth approprate propertes. The rate B` at whch energy leaves the surface ` s derved from the conservaton of energy. Snce the energy exchange between the surfaces s assumed to happen nstantaneously, B` s always gven as the sum of the energy E`, whch s generated by the surface ` tself and the reflecton of the rradatons from the other surfaces. B` = E` + ` B F` (22) B s the rate at whch energy leaves the other surfaces, F` s the fracton of B whch arrves at surface ` and ` s the reflecton coeffcent at surface `. F` depends on the sze and the relatve orentaton of the surfaces ` and. It s also called form factor. We wll not descrbe the meanng of (22) and the calculaton of the form factors n more detal, because these topcs have been dscussed thoroughly n [5, 16] and many other references on heat radaton and computer graphcs. The two basc dfferences between radaton transfer and sound propagaton are dampng of the sound ntensty by the ar, fnte propagaton speed of sound. Snce we are not dealng wth real or vrtual pont sources as n the mage source or ray tracng method, there s no attenuaton wth ncreasng dstance. The dampng consdered here s due to converson of knetc to thermal energy by frcton and molecular effects n the ar. The relaton between the sound pressure p(d) at dstance d from the source and the source pressure p 0 s gven by an exponental law p(d) p 0 = exp(,d) : (23) s the absorpton or extncton coeffcent and depends on the ar temperature and humdty. We may descrbe ths effect concsely by the ar attenuaton factor [15] between the surfaces ` and wth the dstance d` ` = exp(,d`) : (24) ` descrbes the fracton of the rate of energy B F` whch arrves at surface ` n the presence of transmsson losses. Thus the energy balance for sound propagaton n a lossy medum s gven by B` = E` + ` B F` ` : (25)

7 We see that dampng of the sound ntensty by the ar can be modelled by the ar attenuaton factor, whch acts on B n the same way as the form factor F` does. A more fundamental change to the equlbrum equaton (22) s necessary to take the fnte propagaton speed nto account. Equatons (22) and (25) show no tme dependence, snce t s understood that any changes n the system lead nstantaneously to a new equlbrum. No dynamcal or memory effects have to be consdered wth respect to the speed of lght. Ths s no longer true for sound propagaton. The energy transport from dstant surfaces s not only subject to transmsson losses but also to tme delay. Sound energy whch leaves a surface does not arrve n the same tme nstant at other surfaces. To account for the travel tme of the sound waves, we have to formulate the conservaton of energy wth tme dependent quanttes. The tme delays have to be expressed by the dstance between the surfaces and the speed of sound. Ths leads to the followng tme dependent formulaton of the energy balance, whch ncludes propagaton losses and delay B`(t) =E`(t)+`, B t, d` F` ` : (26) c The energy exchange rates B`(t) and the emsson rates E`(t) are now functons of tme. A rough sketch of the algorthm s gven n fg. 9. The nput data are the buldng geometry and the reflecton coeffcents of the surfaces. They serve to calculate the form factors and the ar attenuaton factors. Ths s the most tme consumng task of the whole procedure. However, for a gven geometry, these factors are calculated only once snce they do not change wth tme. Once they are avalable, the energy balance (26) can be solved repeatedly by updatng the tme varable t. We can use the radosty method for the calculaton ofmpulseresponses, fweplace therecever as an addtonal small patch nto the sound feld. Alternatvely, we can also calculate the response to source sgnals at the recever postondrectly. When the recever postonchanges wthtme, then the form factors and ar attenuaton factors from the buldng surfaces to the recever have to be updated as well. All the factors between the buldng surfaces reman unchanged. The source functons enter ths model va the emsson rates E`(t). Snce any surface can act as a source, also movng sources can be easly realzed wthout updatng any factors. 3. SPATIAL HEARING So far, we were only concerned wth sound propagaton n pure techncal terms. The recever was nothng more than a pont n space, where the sound pressure s recorded, e.g. by a mcrophone. However, acoustc renderng s more than just the computer smulaton of buldng geometry surface reflecton coeffcents calculate form factors and ar attenuaton factors solve energy balance up date tme Fgure 9: Algorthm for the radosty method the physcal effect of sound wave propagaton n complex envronments. If we am at a true threedmensonal mpresson of a vrtual sound feld, then also questons of human sound percepton have to be consdered. At ths pont, we leave the secure bass of physcal scence and enter the realm of psychoacoustcs. A tremendous amount of research has been conducted n ths area. For an ntroducton, see [2] and the references contaned theren. We wll quote only some of the common concepts to descrbe what s called spatal hearng. Ths term can be roughly defned by some questons, whch naturally arse, when we try to evaluate the human sound percepton capabltes: How can we tell the drecton of a sound source? How can we tell the dstance of a sound source? How can we separate dfferent sound sources, even n the presence of strong background nose? 3.1. Interaural Tme and Intensty Dfferences The ablty to tell the drecton of a sound source can be traced to two basc dfferences n the sound percepton wth both ears [2]: The nteraural tme dfference (ITD) s the dfference n the arrval tme of a wave front between the two ears. The nteraural ntensty dfference (IID) s the dfference n perceved ntensty between the two ears. Both knds of dfferences act together to provde spatal nformaton about the drecton of arrval of a wave front. The human hearng system can resolve tme dfferences between both ears for sgnals up to frequences of 1 khz. The relaton between the relatve tme dfference of two sgnals and the drecton of arrval s based on a smple geometrcal argument. In ths sense, the ears act as a two-mcrophone-array. For

8 low frequences, there s no notceable ntensty dfference between both ears, due to dffracton at the head. For frequences wth a wavelength shorter than the dameter of the head (typcally 1.5 khz and hgher) dffracton becomes neglgble and an ntensty dfference between both ears s perceved. There s evdence [2], that the human hearng system combnes ITD and IID for the locaton of sound sources. ITD domnates for low frequences, whle IID works for hgh frequences. Applcaton of these concepts to acoustc renderng calls for post processng of the recever sgnal. Once the response to a sound source s determned by themethodsoftheprevoussecton, therecever sgnal has to be splt nto two sgnals for the left and for the rght ear respectvely. These sgnals are delayed and fltered n order to produce the nteraural tme and ntensty dfferences whch correspond to the drecton of the source sgnal Head-Related Transfer Functons The ITD and IID just consdered are only the startng pont for an nterpretaton of the human hearng system. They allow to dscrmnate the source poston n terms of left rght, but they do not explan why we can also locate sound sources above and below or n front or n the back of our head. These effects can only be explaned when the shape of the pnnae s taken nto account. They act as a flter between the sound feld outsde the ears and the eardrum. Techncally, ther effect on spatal hearng can be descrbed by a transfer functon, the so called headrelated transfer functon (HRTF). Of course there are two HRTFs, one for each ear. They are defned as the frequency dependend relaton of the sound pressure at the eardrum to the sound pressure at the recever locaton when the lstener s absent. Much effort has been devoted to measurements of the HRTFs and correspondng data sets for typcal HRTFs are avalable [2]. Ther applcaton to acoustcal renderng s straghtforward. After the recever sgnal has been splt nto a left and rght ear sgnal and both sgnals have been treated wth the approprate ITD and IID nformaton, the fnal step s the flterng wth the correspondng HRTF for each ear. When these sgnals are delvered by ear phones drectly to the lstener's eardrum, they carry the necessary spatal cues to enable spatal hearng of vrtual sound sources. cal smulaton of wave propagaton by a sutable algorthm lke the fnte dfference method. Unfortunately, the numercal expense ncreases wth the fourth power of the hghest sgnal frequency, whch renders ths method sutable for low frequences only. The requred number of floatng pont operatons for the geometrcal methods s ndependent from the sgnal frequency. However, they rely on the assumpton of sound ray propagaton and do not account for dffracton. Ths effect s only neglgble for wavelengths whch are small compared to the room dmensons (roughly above 1 khz). These arguments suggest to splt the source sgnal nto a low frequency and a hgh frequency part. The low frequency part can be modelled by a numercal method, whch consders dffracton. The hgh frequency part can be treated wth a geometrcal method or wth a combnaton of the ray-tracng and the radosty method [10] (see fg. 10). hghpass lowpass ray tracng radosty fnte dff. method Fgure 10: Combnaton of numercal smulaton for low frequences and geometrc smulaton for hgh frequences Fnally, we put the peces together to form a complete acoustc renderng system. We have not mentoned sound modellng, because t s not pecular to our applcaton. Sounds may be prerecorded or syntheszed accordng to some model. The propagaton from the source to the recever s traced wth a system accordng to fg. 10 or a part thereof. The spatal hearng capabltes of a human lstener are taken nto account by splttng the receved sgnal nto two channels for the left and the rght ear, respectvely. Both channels receve the proper delay and flterng to produce the desred psychoacoustc effects (see fg. 11). sound modellng sound prop. h(t), Η(ω) IID, ITD HRTF left rght Fgure 11: Combnaton of sound propagaton and human precepton modellng 4. IMPLEMENTATION Fnally, we wll brefly address some ssues n the practcal mplementaton of the methods and concepts descrbed above. The frst topc s the proper choce of the sound propagaton method. The most general method wth the hghest potental of accuracy s the numer- 5. CONCLUSION As yet, there s no sngle method whch s capable of all aspects of acoustcal renderng n complex envronments as they appear n buldngs. Sound propagaton

9 smulaton by numercal methods wth desktop computers s restrcted to low frequences. The stuaton may change n the future wth the use of more effcent numercal algorthms and the avalablty of more computng power at compettve prces. The mage source method s a straghtforward and computatonal feasble method for very smple room geometres. Advanced geometrc methods lke the ray tracng or the radosty method can be mplemented on modern computers as long as the number of rays n the ray tracng method or the number of patches n the radosty method are chosen reasonably small. However, geometrc methods neglect dffracton effects whch are present for large wavelengths n the order of the room dmensons. Snce numercal and geometrcal methods are complementary, t seems worthwhle to combne both methods. In addton to the smulaton of sound propagaton, also an mpresson of spatal hearng has to be acheved. A proper modellng of the nteraural tme and ntensty dfferences and of the effects of the pnnae by head related transfer functons provde the cues to bnaural hearng. Some aspects of acoustcal renderng have not been consdered. So far, only the effect of sound wave reflecton at walls has been modelled, but no transmsson of sound was taken nto account. Ths would be the prerequste for the assessment of sound absorbng materals and for the determnaton of the total sound level wthn a buldng. Also the queston of accuracy was not rased. Are these real tme methods for acoustc renderng only crude approxmatons to realty or do they have the potental to make quanttatve predctons? Is the sound qualty good enough for a real tme judgment of the propertes of a concert hall or an audtorum? The methods and concepts, whch were revewed n ths paper, provde an exctng extenson to the vsual renderng capabltes of modern desgn programs. Ther usefulness as a desgn tool n ts own rght has yet to be establshed. Not only further laboratory research but also user experence from a growng number of mplementatons wll promote ths process. 6. REFERENCES [1] J. B. Allen and D.A. Berkley. Image method for effcently smulatng small-room acoustcs. J. Acoust. Soc. Am, 65(4): , [2] D. R. Begault. 3-D Sound for Vrtual Realty and Multmeda. Academc Press, Cambrdge, USA, [3] J. Borsh. Extenson of the mage model to arbtrary polyhedra. J. Acoust. Soc. Am, 75(6): , [4] S. M. Dance, J. P. Roberts, and B. M. Sheld. Computer predcton of sound dstrbuton n enclosed spaces usng an nterference pressure model. Appled Acoustcs, 44:53 65, [5] C. M. Goral, K. E. Torrance, D. P. Greenberg, and B. Battale. Modelng the nteracton of lght between dffuse surfaces. Computer Graphcs, 18(3): , [6] J. H. Hahn, H. Fouad, L. Grtz, and J. W. Lee. Integratng sounds and motons n vrtual envronments. Presence: Teleoperators and Vrtual Envrnoments, to appear. [7] F. Henle, R. Rabensten, and A. Stenger. Measurng the lnear and non-lnear propertes of electro-acoustc transmsson systems. In Proc. of Int. Workshop on Acoustc Echo and Nose Control (IWAENC'97), London, to appear. [8] F. Henle and H. W. Schüßler. Measurng the performance of mplemented multrate systems. In Proc. Int. Conf. Acoustcs, Speech and Sgnal Proc. (ICASSP 96), IEEE, [9] S. M. Kuo and D. R. Morgan. Actve Nose Control Systems Algorthms and DSP Implementatons. John Wley & Sons, New York, [10] T. Lewers. A combned beam tracng and radant exchange computer model of room acoustcs. Appled Acoustcs, 38: , [11] A. Quarteron and A. Vall. Numercal Approxmaton of Partal Dfferental Equatons. Sprnger-Verlag, Berln, [12] L. Savoja, J. Backman, A. Järvnen, and T. Takala. Wavegude mesh method for low-frequency smulaton of room acoustcs. In Proc. Int. Congress on Acoustcs (ICA '95), , [13] L. Savoja, T. J. Rnne, and T. Takala. Smulaton of room acoustcs wth a 3-d fnte dfference mesh. In Proc. Int. Computer Musc Conference (ICMC '94), , [14] H. W. Schüßler and F. Henle. Measurng the propertes of mplemented dgtal systems. FRE- QUENZ, 48(3-7), [15] J. Sh, A. Zhang, J. Encarnacao, and M. Göbel. A modfed radosty algorthm for ntegrated vsual and audtory renderng. Computers & Graphcs, 17(6): , [16] R. Segel and J. R. Howell. Thermal Radaton Heat Transfer. Hemsphere Publshng Corporaton, Washngton, [17] L. J. Zomek. Fundamentals of Acoustc Feld Theory and Space-Tme Sgnal Processng. CRC Press, Boca Raton, 1995.