Convention Paper Presented at the 124th Convention 2008 May Amsterdam, The Netherlands

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1 Audio Engineering Soiety Convention Paper Presented at the 24th Convention 2008 May 7 20 Amsterdam, The Netherlands The papers at this Convention have been seleted on the basis of a submitted abstrat and extended preis that have been peer reviewed by at least two qualified anonymous reviewers. This onvention paper has been reprodued from the author s advane manusript, without editing, orretions, or onsideration by the Review Board. The AES takes no responsibility for the ontents. Additional papers may be obtained by sending request and remittane to Audio Engineering Soiety, 60 East 42 nd Street, New York, New York , USA; also see All rights reserved. Reprodution of this paper, or any portion thereof, is not permitted without diret permission from the Journal of the Audio Engineering Soiety. Sasha Spors, Rudolf Rabenstein 2, and Jens Ahrens Deutshe Telekom Laboratories, Berlin University of Tehnology, Ernst-Reuter-Platz 7, 0587 Berlin, Germany 2 Multimedia and Signal Proessing, University of Erlangen-Nuremberg, Cauerstrasse 7, 9058 Erlangen, Germany Correspondene should be addressed to Sasha Spors (Sasha.Spors@telekom.de) ABSTRACT Wave field synthesis is a spatial sound field reprodution tehnique aiming at authenti reprodution of auditory senes. Its theoretial foundation has been developed almost 20 years ago and has been improved onsiderably sine then. Most of the original work on wave field synthesis is restrited to the reprodution in a planar listening area using linear loudspeaker arrays. Extensions like arbitrarily shaped distributions of seondary soures and three-dimensional reprodution in a listening volume have not been disussed in a unified framework so far. This paper revisits the theory of wave field synthesis and presents a unified theoretial framework overing arbitrarily shaped loudspeaker arrays for two- and three-dimensional reprodution. The paper additionally gives an overview on the artifats resulting in pratial setups and briefly disusses some extensions to the traditional onepts of WFS.. INTRODUCTION Wave field synthesis (WFS) is a spatial sound field reprodution tehnique that utilizes a high number of loudspeakers to reate a virtual auditory sene over a large listening area. It overomes some of the limitations of stereophoni reprodution tehniques, like e. g. the sweet-spot. A first onept, of what is nowadays known as WFS, was presented by Snow et al. [] more than 50 years ago. However, tehnial onstraints prohibited the employment of a high number of loudspeakers for sound reprodution. The authors therefore employed only some few loudspeakers and essentially laid the theoretial fundament for stereophoni tehniques. It took quite some time until the initial ideas of Snow have been taken up again. The theoretial framework of WFS was initially formulated by Berkhout et al. at the Delft University of Tehnology almost 20 years ago [2]. However, it seems that the term wave field synthesis has been mentioned the first time some years later [3]. Also around that time first laboratory setups of WFS sys-

2 tems have been realized. Sound reprodution using WFS has gained quite some attration in the spatial audio researh ommunity. Besides various researh projets, like e. g. the EC IST funded projet CAR- ROUSO [4], also a number of PhD theses have been written in the ontext of WFS so far [5, 6, 7, 8, 9, 0,, 2, 3, 4]. General overviews on WFS an be found e. g. in [5, 6, 7, 8, 9, 20, 2, 22, 23, 24]. Most of the original work on WFS onsiders the reprodution in a planar listening area (twodimensional wave field synthesis), using a linear distribution of loudspeakers. Although the theory of WFS has been extended in various aspets, topis like arbitrarily shaped distributions of loudspeakers and three-dimensional reprodution have gained only little attention so far. This paper will revisit the physial bakground of WFS and will present a unified framework that overs these aspets. For this purpose our paper begins in Setion 2 with the generi formulation of the underlying physial problem using the Kirhhoff-Helmholtz integral. It will be shown how to derive a formulation from this fundamental priniple that is appliable in pratial setups. Setion 3 speializes the generi theory of WFS developed so far, to the ase of threedimensional reprodution in a volume. The threedimensional formulation serves as basis for the desription of the onventional two-dimensional WFS shemes introdued in Setion 4. In order to span the bridge to the traditional theory of WFS, the loudspeaker driving funtions derived within this paper will be ompared to the lassial WFS literature. Besides revisiting the foundations of WFS, Setion 5 will give an overview on the artifats resulting from further assumptions and simplifiations performed in pratial setups. Finally some extensions to WFS will be disussed in Setion 6. The following onventions are used throughout this paper: For salar variables lower ase denotes the time domain, upper ase the temporal frequeny domain. The temporal frequeny variable is denoted by ω = 2πf. Vetors are denoted by lower ase boldfae. The three-dimensional position vetor in Cartesian oordinates is given as x = [x y z] T. Twodimensional wave fields are also onsidered within this paper. The required redution in dimensionality is performed by assuming that the reprodued wave field is independent from the z-oordinate, e. g. P(x, y, z, ω) = P(x, y, ω). 2. BASIC THEORY This setion introdues the basi theory of wave field synthesis. 2.. The Kirhhoff-Helmholtz Integral A loudspeaker system surrounding the listener an be regarded as an inhomogeneous boundary ondition for the wave equation. This will be illustrated in the following. The solution of the homogeneous wave equation for a bounded region V with respet to inhomogeneous boundary onditions is given by the Kirhhoff- Helmholtz integral [25] ( P(x, ω) = G(x x 0, ω) V n P(x 0, ω) P(x 0, ω) ) n G(x x 0, ω) ds 0, () where P(x, ω) denotes the pressure field inside a bounded region V enlosed by the boundary V (x V ), G(x x 0, ω) a suitable hosen Green s funtion, P(x 0, ω) the aousti pressure at the boundary V (x 0 V ) and n the inward pointing normal vetor of V. The abbreviation n denotes the diretional gradient in diretion of the normal vetor n. For instane n P(x 0, ω) is n P(x 0, ω) = P(x, ω),n(x 0 ), (2) x=x0 where, denotes the salar produt of two vetors. The wave field P(x, ω) outside of V is zero and V is assumed to be soure-free. Figure illustrates the geometry. The Green s funtion G(x x 0, ω) represents the solution of the inhomogeneous wave equation for exitation with a spatio-temporal Dira pulse at the position x 0. It has to fulfill the homogeneous boundary onditions imposed on V. For sound reprodution typially free-field propagation within V is assumed. This means that V is free of any objets and that the boundary V does not restrit propagation. The Green s funtion is then given as the free-field solution of the wave equation and is referred to as free-field Green s funtion G 0 (x x 0, ω). The free-field Green s funtion an be interpreted as the spatio-temporal transfer funtion of a monopole plaed at the point x 0 and its diretional gradient as the spatio-temporal transfer funtion of a dipole Page 2 of 9

3 virtual soure S(x, ω) x 0 n 0 V P(x, ω) x V Fig. : Illustration of the geometry used for the Kirhhoff-Helmholtz integral (). at the point x 0, whose main axis points towards n. Equation () states that the wave field P(x, ω) inside V is fully determined by the pressure P(x, ω) and its diretional gradient on the boundary V. If the Green s funtion is realized by a ontinuous distribution of appropriately driven monopole and dipole soures whih are plaed on the boundary V, the wave field within V is fully determined by these soures. This priniple an be used for sound reprodution as will be illustrated in the following. In this ontext the monopole and dipole soures on the boundary are referred to as (monopole/dipole) seondary soures. For authenti sound field reprodution it is desired to reprodue the wave field S(x, ω) of a virtual soure inside a limited area (listening area) as losely as possible. In the following, the listening area is assumed to be the bounded region V (see Fig. ). Conluding the onsiderations given so far, authenti sound field reprodution an be realized if a distribution of seondary monopole and dipole soures on the boundary V of the listening area V are driven by the diretional gradient and the pressure of the wave field of the virtual soure S(x, ω), respetively. Thus P(x 0, ω) in Eq. () is given by the values of S(x, ω) on V. The wave field P(x, ω) inside the listening area V is then equal to the wave field S(x, ω) of the virtual soure. The Kirhhoff-Helmholtz integral and its interpretation given above lay the theoretial foundation for WFS and other massive multihannel sound reprodution systems. It is desirable for a pratial implementation to disard one of the two types of seondary soures that the Kirhhoff-Helmholtz integral employs. Typially the dipole soures are removed, sine monopole soures an be realized reasonably well by loudspeakers with losed abinets. Aording to [25] two different tehniques exist to derive monopole only versions of the Kirhhoff-Helmholtz integral: the simple soure approah and a modifiation of the free-field Green s funtion used in the Kirhhoff- Helmholtz integral. These two approahes will be disussed in the following two subsetions Simple Soure Approah The dipole seondary soures in the Kirhhoff- Helmholtz integral an be disarded by onsidering two equivalent but spatially disjunt problems. The simple soure approah is derived by onstruting an exterior and separately an interior problem with respet to the boundary V and linking both problems by requiring that the pressure is ontinuous and the diretional gradient is disontinuous at the boundary V [25]. This proedure results in P(x, ω) = µ(x 0, ω)g 0 (x x 0, ω) ds 0. (3) V Equation (3) states that a distribution of monopole soures on V driven by µ(x 0, ω) fully determines the wave field P(x, ω) within and outside of V. Note, that ontrary to the Kirhhoff-Helmholtz formulation the wave field outside of V will not be zero in this ase. It is stated in [25] that the simple soure approah delivers the same results as the Kirhhoff-Helmholtz formulation when onsidering either an interior or exterior problem. Sound reprodution an be regarded as interior problem. The soure strength µ(x 0, ω) is given by the underlying physial problem whih satisfies the stated boundary onditions. For authenti sound field reprodution it is required that the field P(x, ω) within V is equal to the wave field of the virtual soure S(x, ω). The appropriate seondary soure strength µ(x 0, ω) an be derived by onstruting an exterior field that satisfies the required boundary onditions. In general this will only be possible by onsidering speial geometries of the seondary soure ontour V. In higher-order Ambisonis [26, 27], and other reprodution tehniques [28, 29, 30] whih are inherently Page 3 of 9

4 based on the simple soure approah, Eq. (3) is expliitly solved with respet to µ(x 0, ω). This is typially performed by expanding the respetive wave fields using orthogonal wave field expansions. The simple soure formulation (3) ensures that a unique solution for the seondary soure strength µ(x 0, ω) exists Elimination of Dipole Seondary Soures in WFS The seond term in the Kirhhoff-Helmholtz integral (), representing the dipole seondary soures, an be disarded by modifying the Green s funtion used in the Kirhhoff-Helmholtz integral [25]. The modified Green s funtion G N (x x 0, ω) has to obey the following ondition n G N(x x 0, ω) = 0, (4) x0 V in order to eliminate the dipole seondary soures. Condition (4) formulates a homogeneous Neumann boundary ondition imposed on V. The modified Green s funtion is typially termed Neumann Green s funtion. As a onsequene of the ondition given by Eq. (4), the boundary V will be impliitly modeled as an aoustially rigid surfae for the seondary soures. The desired Neumann Green s funtion G N (x x 0, ω) an be derived by adding a suitable homogeneous solution (with respet to the region V ) to the free-field Green s funtion G 0 (x x 0, ω). The expliit form of the Neumann Green s funtion depends on the geometry of the boundary V. A losed form solution an only be found for rather simple geometries like spheres and planar boundaries. Additionally, the suh derived Neumann Green s funtion has to be realized by physially existing seondary soures. Depending on the expliit form of the Neumann Green s funtion suh seondary soures may be hard to realize in pratie. In the ontext of WFS linear seondary soure ontours have been onsidered mainly so far. A suitable Neumann Green s funtion for a planar/linear boundary V is given by [25] G N (x x 0, ω) = G 0 (x x 0, ω)+g 0 (x m (x) x 0, ω). (5) A solution fulfilling Eq. (4) is given by hoosing the reeiver point x m (x) as the point x mirrored at the planar boundary V at the position x 0. Note, that due to the speialized geometry x x 0 = x m x 0 and thus [24] G N (x x 0, ω) = 2 G 0 (x x 0, ω). (6) Hene, in this speial ase G N (x x 0, ω) is equal to a point soure with double strength. Introduing G N (x x 0, ω) into the Kirhhoff- Helmholtz integral for a planar geometry derives the first Rayleigh integral, whih is the basis for the traditional derivation of WFS [2, 5, 6, 7]. However, this theoretial basis holds only for linear/planar seondary soure distributions. In the following an extension to arbitrarily shaped seondary soure ontours will be developed Extension to Arbitrarily Shaped Contours It is assumed that Eq. (5) holds also approximately for other geometries. In this ase the reeiver point x m is hosen as the point x mirrored at the tangent to the boundary V at the position x 0. The elimination of the seondary dipole soures for an arbitrary seondary soure ontour V has two onsequenes:. the wave field outside of V will not be zero, and 2. the reprodued wave field will not math the virtual soure field exatly within V. The first onsequene implies that the boundary V has to be onvex, so that no ontributions from the wave field outside of the listening area V propagate bak into the listening area. The seond is a onsequene of approximating the Neumann Green s funtion for arbitrary geometries using the solution given by Eq. (5). As mentioned before, the boundary V will be impliitly modeled as rigid boundary. The appropriate Neumann Green s funtion for a partiular geometry ompensates inherently for these refletions. Using the Neumann Green s funtion Eq. (5) for bend seondary soure ontours leads to artifats in the reprodued wave field. The most prominent are that the reprodution of the desired wave field will be superimposed by undesired refletions. These artifats an be attenuated by a modifiation of the driving funtion, as will be illustrated in the following. The main energy of the undesired refletions is reprodued by those seondary soures where the loal propagation diretion of the virtual wave field does Page 4 of 9

5 not oinide with the normal vetor n of the seondary soure. Sine we are free to hoose the seondary soures used for reprodution, these undesired refletions an be attenuated by muting those seondary soures whih reprodue the refletions. Following this onept, the reprodued wave field reads P(x, ω) = 2a(x 0 ) n S(x 0, ω)g 0 (x x 0, ω) ds 0, (7) V where a(x 0 ) denotes a suitably hosen window funtion. This funtion takes are that only those seondary soures are ative where the loal propagation diretion of the virtual soure at the position x 0 has a positive omponent in diretion of the normal vetor n of the seondary soure at that position. It was proposed in [3, 32] to formulate this ondition analytially on basis of the aousti intensity vetor {, if ĪS(x 0, ω),n(x 0 ) > 0, a(x 0 ) = (8) 0, otherwise. The time averaged aousti intensity vetor ĪS(x, ω) for the wave field of the virtual soure is defined as Ī S (x, ω) = 2 R{S(x, ω)v S(x, ω) }, (9) where V S (x, ω) denotes the partile veloity field of the virtual soure S(x, ω), R{ } the real part of its argument and the supersript the onjugate omplex of a variable. The Green s funtion in Eq. (7) haraterizes the field of the seondary soures, the remaining terms their strength. The strength will be termed as seondary soure driving funtion in the following. The seondary soure driving funtion D(x 0, ω) is given as D(x 0, ω) = 2a(x 0 ) n S(x 0, ω). (0) The seondary soure driving funtion plays an important role sine it determines the loudspeaker signals in a pratial implementation. It is the basis for three- and two-dimensional WFS approahes disussed within the sope of this paper. Summarizing the results of this setion, the sound pressure P(x, ω) inside the listening area an be expressed by the seondary soure driving funtions D(x 0, ω) and the Green s funtions G(x x 0, ω) of the monopoles at the boundary V as P(x, ω) = D(x 0, ω)g 0 (x x 0, ω) ds 0. () V For an arbitrarily shaped boundary V the reprodued wave field will not exatly math the virtual soure field S(x, ω) within V. However, pratie has revealed that the assumptions used to derive the driving funtion provide a reasonable approximation for sound reprodution purposes. A detailed analysis of the resulting artifats is an open researh topi Virtual Soure Models Model-based rendering of a virtual soures requires appropriate models for their wave fields. This setion expliitly introdues two ommonly used virtual soure models, plane and spherial waves. It is also shown how to determine the window funtion a(x 0 ) from (0) for these two speial ases. The wave field of a plane wave is given as S pw (x, ω) = Ŝpw(ω) e j ω nt pw x, (2) where n pw denotes the propagation diretion of the plane wave and Ŝpw(ω) its spetrum. The window funtion a pw (x 0 ) for a virtual plane wave an be derived by evaluating (9) for the pressure S pw (x, ω) and veloity field V pw (x, ω) of a plane wave as [3] {, if n pw,n(x 0 ) > 0, a pw (x 0 ) = (3) 0, otherwise. The wave field of a spherial wave with stationary position is given as S sw (x, ω) = Ŝsw(ω) e j ω x xs x x S, (4) where x S denotes the enter position of the spherial wave and Ŝsw(ω) its spetrum in radial diretion. The window funtion a sw (x 0 ) for a virtual spherial wave is given as [3] {, if x 0 x S,n(x 0 ) > 0, a sw (x 0 ) = (5) 0, otherwise. Besides these two basi types of virtual soures also other models have been developed in the past. Page 5 of 9

6 Models for omplex soures and appropriate driving funtions for WFS have been developed by e. g. [33, 34, 35, 36]. However, most of this work is based upon the assumption of a linear seondary soure distribution whih requires no sensible seletion of the ative seondary soures. Following the generalization of the linear ase to urved and losed seondary soure ontours, given in Setion 2.3, the aousti intensity vetor of the omplex soure an be used to selet the ative seondary soures in the general ase. The model of a spherial wave given by Eq. (4) is based on the assumption of a stationary soure position. Most of the urrent implementations of WFS systems render moving soures as a sequene of stationary soure positions that hange over time [37]. Reently this situation has been improved by expliitly applying models of moving soures [38]. The underlying theory of WFS assumes that the listening area V is free of soures. Consequently no virtual soure S(x, ω) an be plaed within the listening area in order to determine an appropriate driving funtion for that situation. However, it is possible to reprodue a point soure (with some restritions) in the listening area by applying the timereversal priniple [39, 40]. Suh soures are termed as foused soures. Reently, the theory of foused soures has been extended to allow the reprodution of diretional foused soures [4]. Note, that the proposed seondary soure seletion sheme (8) has to be modified to over foused soures [3]. 3. THREE-DIMENSIONAL WAVE FIELD SYN- THESIS The generi theory of WFS developed in Setion 2 holds for two- and three-dimensional WFS systems. This setion will speialize the theory to the ase of three-dimensional reprodution in a listening volume. The Green s funtion used in the reprodution equation () determines the harateristis of the seondary soures. The speifi form of the freefield Green s funtion depends on the dimensionality of the problem. The three-dimensional free-field Green s funtion is given as [25] G 0,3D (x x 0, ω) = 4π e j ω x x0. (6) x x S Equation (6) an be interpreted as the field of a point soure with monopole harateristis loated at the position x Arbitrarily Shaped Seondary Soure Contours Three-dimensional WFS an be realized by surrounding the listening volume V by a ontinuous distribution of point soures plaed on the boundary V. These seondary soures are driven by the seondary soure driving funtion (0). The driving funtion is given by the diretional gradient of the virtual soure wave field and the window funtion a(x 0 ). Hene, the expliit form of the driving funtion depends on the virtual soure and the geometry of the sound reprodution system. The driving funtion for a plane wave is determined by the diretion gradient of the wave field of a plane wave (2) and the window funtion (3) as D pw,3d (x 0, ω) = 2a pw (x 0 ) nt pw n(x 0) jωŝpw(ω)e j ω nt pw x0. (7) A time-domain version of the driving funtion (7) is useful to derive an effiient implementation of WFS. Inverse Fourier transformation of Eq. (7) reveals the time-domain version of the driving signal d pw,3d (x 0, t) = 2a pw (x 0 ) nt pw n(x 0) d dtŝpw(t nt pw x 0 ), (8) where the differentiation theorem of the Fourier transformation was used. Equation (8) states that the driving signal for a plane wave an be omputed effiiently in the time-domain by weighting the derivative of the time-shifted soure signal ŝ pw (t). However, the differentiation of the virtual soure signal may also be performed by filtering the signal by a filter with jω-harateristi. This is espeially useful when onsidering the effets of spatial aliasing (see Setion 5.). The driving funtion for a virtual spherial wave an be derived by following the same proedure as outline above for a virtual plane wave. In the frequeny Page 6 of 9

7 domain it is given as D sw,3d (x 0, ω) = 2a sw (x 0 ) (x 0 x S ) T n(x 0 ) x 0 x S 2 ( x 0 x S + jω ) Ŝ sw (ω)e j ω x0 xs. (9) The time-domain version of the driving signal for a spherial wave is derived by inverse Fourier transformation of Eq. (9) d sw,3d (x 0, t) = 2a sw (x 0 ) (x 0 x S ) T n(x 0 ) x 0 x S 2 ( x 0 x S + ) d ŝ sw (t x 0 x S ). (20) dt For a spherial wave, the driving signal in the timedomain is given by a weighted linear superposition of the time-shifted soure signal ŝ sw (t) and its derivative. Pratial implementations of three-dimensional sound reprodution systems typially exhibit spherial or uboid shape. The detailed disussion of spherial WFS systems is out of sope in this paper. A planar seondary soure distribution is the basi building blok of a uboid shaped reprodution system. A planar distribution will be disussed in detail in the next setion Planar Seondary Soure Distribution The losed ontour integral () over the surfae V an be degenerated to an integral over an infinite plane. In brief, this degeneration is ahieved by splitting the losed ontour V into a planar boundary and a half-sphere. The integration over the halfsphere an be omitted by applying the Sommerfeld radiation ondition [25]. It will be assumed in the following, without loss of generality, that the seondary soure distribution is loated on the xz-plane at y = 0. Other ases an be regarded as simple translation or rotation of this speial ase. The reprodued wave field for a planar distribution of seondary point soures on the xz-plane is given as P(x, ω) = D 3D (x 0, ω)g 0,3D (x x 0, ω) dx 0 dz 0, (2) with x 0 = [x 0 0 z 0 ] T. Equation (2) is known as the first Rayleigh integral. The reprodued wave field P(x, ω) will be mirrored at the seondary soure distribution as a onsequene of the Neumann boundary ondition (4). Hene, the reprodued wave field is only orret in one of the two half-volumes separated by the seondary soure distribution. The diretion of the normal vetor n speifies the onsidered half-volume. We will onsider the half-volume with y 0 as the listening area in the sequel. The normal vetor for this ase is given as n = [0 0] T. The seondary soure driving funtion for a virtual plane wave is given by speializing Eq. (7). Due to the symmetry of the reprodued wave field with respet to the seondary soure distribution it is reasonable to limit the inidene angle of the virtual plane wave to the ase n y,pw > 0, hene to plane waves traveling into the positive y-diretion. Aordingly to Eq. (3), the value of the window funtion for seletion of ative seondary soures will be a pw (x 0 ) =. The driving funtion for a virtual spherial wave is given by speializing Eq. (9). Due to the symmetry of the reprodued wave field with respet to the seondary soure distribution it is reasonable to onstrain the possible positions of the point soure to y S < 0. Aordingly to Eq. (5), the value of the window funtion for seletion of ative seondary soures will be a sw (x 0 ) = in this ase. Up to now, the seondary soure distribution was assumed to be of infinite size and ontinuous. Pratial implementations of three-dimensional planar WFS systems will onsist of a limited number of seondary soures plaed at disrete positions. Sine loudspeakers will be used as seondary soures in pratie, these disrete distributions of seondary soures are termed as (planar) loudspeaker arrays. Two types of artifats may emerge from the spatial trunation and disretization: () trunation and (2) spatial aliasing artifats. Trunation artifats an be analyzed by multiplying the driving funtion with a window modeling the aperture of the loudspeaker array, spatial sampling by multiplying the driving funtion with a series of spatial Dira pulses. A detailed analysis of both artifats is beyond the sope of this paper. However, both have been analyzed already for linear loudspeaker arrays [42]. The results derived here an be applied straightforwardly to planar loudspeakers arrays due to the separability Page 7 of 9

8 of the Cartesian oordinate system. A summary of aliasing, trunation and other artifats of WFS an be found in Setion 5. The reprodued wave field for a planar ontinuous distribution of infinite size will exatly math the wave field of the virtual soure within the listening area. This an be proven by inserting the driving funtions into the reprodution equation (2). Artifats will our for other geometries of the seondary soure distribution. This is due to the fat that the derived Neumann Green s funtion only fulfills the required Neumann boundary ondition exatly in this speial ase Example for Planar Seondary Soure Distribution Figure 2 illustrates the wave field reprodued by a planar loudspeaker array in the plane z = 0. The loudspeaker array onsist of point soures with a distane of x = y = 0.5 m between them. The rather high number of loudspeakers has been hosen to avoid signifiant aperture and aliasing artifats in the illustrated region. Figure 2(a) shows the reprodued wave field when using the plane wave driving funtion (7) for the reprodution of a monohromati virtual plane wave with frequeny f pw = 500 Hz and propagation diretion n pw = [0 0] T. Aordingly to [42] spatial aliasing will only our for frequenies above 2 khz for this onfiguration. Figure 2(b) shows the reprodued wave field when using the spherial wave driving funtion (9) for the reprodution of a monohromati virtual spherial wave with frequeny f sw = 500 Hz and position x S = [0 2 0] T m. It an be seen learly that both wave fields are reprodued aurately by the planar distribution of seondary point soures. 4. TWO-DIMENSIONAL WAVE FIELD SYN- THESIS The tehnial realization of a three-dimensional WFS system would involve a very high number of loudspeakers and reprodution hannels. The majority of WFS systems realized so far are therefore restrited to the reprodution in a plane only. This redution of dimensionality is reasonable for most senarios due to the spatial harateristis of human hearing. Preferably this listening plane should be leveled with the listeners ears. Suh systems will be termed as two-dimensional WFS systems in the following. The redution in dimensionality has one major drawbak: two-dimensional WFS systems are not apable to reprodue wave fields whih have ontributions emerging from soures above or below the plane where the loudspeakers are ontained in. However, this restrition holds also for most of the urrently applied surround systems. Note, that in the following all position vetors are assumed to be twodimensional. 4.. Line Soures as Seondary Soures The generi theory of WFS developed in Setion 2 an be speialized to two-dimensional reprodution by using the two-dimensional free-field Green s funtion as wave field for the seondary soures. The two-dimensional free-field Green s funtion is given as [25] G 2D (x x 0, ω) = j 4 H(2) 0 (ω x x 0 ), (22) where H (2) 0 ( ) denotes the zeroth-th order Hankel funtion of seond kind [43]. For the definition of a two-dimensional wave field used within the ontext of this paper, Eq. (22) an be interpreted as the field of a line soure. This line soure is loated parallel to the z-axis and intersets with the reprodution plane at the position x 0. Loudspeakers that approximately have the properties of line soures are hardly available. For instane suh a loudspeaker should have infinite length whih is not realizable. Hene, using line soures as seondary soures for WFS serves more as theoretial framework to derive various properties of WFS and for illustration purposes. Therefore, this senario is only disussed in brief here. Note, that in a truly two-dimensional wave propagation senario, Eq. (22) represents the wave field produed by a spatio-temporal Dira pulse at the position x 0. The driving funtion for virtual plane wave, as derived in Setion 3., is independent from the underlying dimensionality of the wave fields. Hene, Eq. (7) holds also for two-dimensional WFS (with n z,pw = 0) using line soures as seondary soures. The driving funtion for a virtual plane wave an be derived straightforwardly from Eq. (7) by using the two-dimensional normal vetor n pw = [osθ pw sin θ pw ] T where θ pw denotes the inidene angle of the plane wave. Page 8 of 9

9 y > [m].5 y > [m] x > [m] (a) plane wave (f pw = 500 Hz, n pw = [00] T ) x > [m] (b) spherial wave (f sw = 500 Hz, x S = [0 2 0] T m) Fig. 2: Wave fields reprodued in the plane z = 0 by a planar seondary soure distribution of point soures with a distane of x = y = 0.5 m between them. In two-dimensional wave propagation the wave field of a line soure is given by Eq. (22). The orresponding virtual wave field will be termed as virtual ylindrial wave in the following. The driving funtion for a virtual ylindrial wave plaed at the position x S an be derived as D y,2d (x 0, ω) = 2 a y(x 0 ) (x 0 x S ) T n(x 0 ) x 0 x S jω Ŝy(ω)H (2) (ω x 0 x S ), (23) where a y = a sw due to the radial symmetry of both line soures and point soures. Figure 3 illustrates the wave field reprodued by a linear distribution of line soures. The loudspeaker array onsists of 00 line soures with a distane of x = 0.5 m between them. Figure 3(a) shows the reprodued wave field when using the plane wave driving funtion (7), Fig. 3(b) shows the reprodued wave field when using the ylindrial wave driving funtion (23). It an be seen learly that both wave fields are reprodued aurately by the linear distribution of seondary line soures. Please note, that using the driving funtion (9) of a spherial wave for two-dimensional WFS will result in artifats in the reprodued wave field, most notably an inorret amplitude deay Point Soures as Seondary Soures It has been shown in the previous setion that line soures are the appropriate hoie as seondary soures for two-dimensional WFS. However, it is desirable to use loudspeakers with losed abinets as seondary soures, due to the fat that suh loudspeakers are widely available. These approximately have the harateristis of an aousti point soure. In order to analyze and ompensate the error introdued by this seondary soure type mismath a loser look is taken at the properties of point and line soures. The asymptoti expansion of the Hankel funtions for large arguments [43] is used to approximate the two-dimensional Green s funtion G 2D (x x 0, ω) as follows G 2D (x x 0, ω) 2π x x 0 j ω e j ω x x0 4π x x 0 }{{} G 3D(x x 0,ω) (24) Comparing the right fration of Eq. (24) with Eq. (6) reveals that the given approximation of G 2D (x x 0, ω) is equivalent to G 3D (x x 0, ω) when applying a spetral and amplitude orretion. Note that the orretions outlined above assume that the large argument approximation ( ω x x 0 ) holds. For low frequenies and positions lose to the. Page 9 of 9

10 y > [m].5 y > [m] x > [m] x > [m] (a) plane wave (f pw = 500 Hz, n pw = [0] T ) (b) ylindrial wave (f y = 500 Hz, x S = [0 2] T m) Fig. 3: Wave fields reprodued by a linear distribution of 00 seondary line soures with a distane of x = 0.5 m between them. seondary soures this approximation might be inaurate. The large argument approximation of the Hankel funtion is also known as far-field or stationary phase approximation. Introduing the approximation (24) of the twodimensional free-field Green s funtion into Eq. () yields P(x, ω) = 2π x x 0 V j ω D 2D (x 0, ω)g 3D (x x 0, ω) ds 0. (25) The ompensation of the seondary soure type mismath an be inluded into the driving funtion used for two-dimensional WFS with point soures as seondary soures. It an be seen from Eq. (25) that the required spetral orretion is independent from the reeiver position x. However, the amplitude orretion depends on the reeiver position. As a onsequene, the amplitude an only be orreted for one reeiver position in the listening area. For other positions, amplitude errors will be present. The position that is used for the amplitude orretion is denoted by x ref in the following. The orreted driving funtion is then given as D 2.5D (x 0, ω) = 2π xref x 0 D 2D (x 0, ω), j ω (26) where the notation 2.5D aounts for the mixture of two dimensional reprodution using point soures as seondary soures. The driving funtion for a virtual plane wave an be derived straightforwardly by introduing Eq. (7) into Eq. (26) as D pw,2.5d (x 0, ω) = 2a pw (x 0 ) 2π x ref x 0 j ω Ŝpw(ω)n T pw n(x 0)e j ω nt pw x0. (27) Inverse Fourier transformation of Eq. (27) reveals the time-domain driving funtion d pw,2.5d (x 0, t) = w pw δ(t nt pw x 0 ) (f pw (t) ŝ pw (t)), (28) where all weighting terms have been ombined into w pw = 2a pw (x 0 ) 2π x ref x 0 n T pwn(x 0 ) and f pw (t) is given by the inverse Fourier transformation of j ω. Equation (28) states that the driving signal Page 0 of 9

11 for a virtual plane wave an be omputed effiiently in the time-domain by weighting and delaying the pre-filtered soure signal ŝ pw (t). Note, that the prefiltering is independent from the seondary soure position. Hene, it only has to be performed one for all seondary soures in advane to the seondary soure dependent weighting and delaying. It is interesting to note that the pre-filtering by f pw (t) an also be understood as taking the half-derivative of the soure signal ŝ pw (t). The driving funtion for a ylindrial wave was derived in Setion 4.. Sine, we are aiming at twodimensional WFS this would be the appropriate driving funtion at first sight. However, line soures don t exhibit a flat frequeny response as an be seen from (22). In oder to overome this drawbak, point soure models are typially used as virtual soure model in two-dimensional WFS with point soures as seondary soures. The driving funtion for a virtual spherial wave is given by introduing the driving funtion for a spherial wave (9) into Eq. (26) D sw,2.5d (x 0, ω) = 2a sw (x 0 ) (x 0 x S ) T n(x 0 ) 2π xref x 0 x 0 x S ( ) jω j ω x 0 x S + Ŝ sw (ω) e j ω x0 xs. x 0 x S (29) The time-domain driving funtion, derived by inverse Fourier transformation of Eq. (29), reads d sw,2.5d (x 0, t) = w sw δ(t x 0 x S ) (f sw (t) ŝ sw (t)), (30) where all weighting fators have been olleted in w sw. The pre-filter f sw (t) is defined as follows {( )} f sw (t) = F jω j ω x 0 x S +, (3) where F denotes the inverse Fourier transformation. As for the plane wave, the driving signal for a spherial wave an be omputed effiiently in the time domain by weighting and delaying the prefiltered soure signal. Figure 4 illustrates the wave field reprodued by a irular distribution of point soures. The loudspeaker array onsist of 56 equiangular positioned point soures with a radius of R =.50 m. Figure 4(a) shows the reprodued wave field when using the plane wave driving funtion (27), Fig. 4(b) shows the reprodued wave field when using the spherial wave driving funtion (29). The seondary soure seletion riterion is indiated by the solid loudspeaker symbols. The amplitude errors inherent to the use of seondary point soures for two-dimensional reprodution an be seen learly in Fig. 4(a). A plane wave is supposed not to lose amplitude over distane Linear Distributions of Seondary Point Soures The wave field reprodued by a linear distribution of seondary point soures is given by speializing Eq. (25) to a linear geometry P(x, ω) = D 2.5D (x 0, ω)g 3D (x x 0, ω) dx 0, (32) where it is assumed that the seondary soure distribution is loated on the x-axis. The integral (32) is also known as the 2.5-dimensional Rayleigh integral [6, 5]. It was shown in [29] that a linear distribution of seondary point soures is apable of reproduing the amplitude orretly only on a line parallel to the seondary soure distribution. This was also shown in the traditional WFS literature within the limitations of the stationary phase approximation. Hene, the amplitude orretion by 2π x ref x 0 in Eq. (26) an be assumed to be onstant for the linear ase. The driving funtions for plane and spherial waves that have been derived in the previous setion an be applied straightforwardly to the linear ase. The reprodued wave field will be symmetrial to the x- axis. Hene, the inidene angle of the virtual plane wave has to be restrited to n y,pw > 0 and the position of the virtual point soure to y S < 0. As for the planar seondary soure distribution disussed in Setion 3.2 no expliit seondary soure seletion is required in this ase. Figure 5 illustrates the wave field reprodued by a linear distribution of seondary point soures for a virtual plane wave and spherial wave. The Page of 9

12 y > [m] y > [m] x > [m] (a) plane wave (f pw = 500 Hz, n pw = [0] T ) x > [m] (b) spherial wave (f sw = 500 Hz, x S = [0 2] T m) Fig. 4: Wave fields reprodued by a irular distribution of 56 point soures with a radius of R =.50 m. The ative seondary soures are indiated by the solid loudspeaker symbols y > [m].5 y > [m] x > [m] x > [m] (a) plane wave (f pw = 500 Hz, n pw = [0] T ) (b) spherial wave (f sw = 500 Hz, x S = [0 2] T m) Fig. 5: Wave fields reprodued by a linear distribution of 00 seondary point soures with a distane of x = 0.5 m between them. Page 2 of 9

13 wave field for a virtual plane wave is shown in Fig. 5(a), the wave field for a virtual spherial wave in Fig. 5(b). The amplitude errors inherent to the use of seondary point soures in this situation an be seen learly in Fig. 5(a) Comparison to the Traditional Formulation of WFS In this setion, we provide a link between the traditional WFS formulation [6] and the proposed one. As denoted in Setion 2.3, the traditional WFS literature has onentrated on linear distributions of seondary point soures reproduing a monopole point soure. We will therefore exemplarily onsider this speial ase. The traditional WFS formulation of the driving funtion D trad (x 0, ω) for a linear distribution of seondary point soures parallel to the x-axis at y = y 0 reproduing a monopole point soure at position x S reads [6] j ω D trad (x 0, ω) = Ŝ(ω) 2π x ref x 0 x S x 0 + x ref x 0 osφ e j ω xs x0 xs x 0, (33) whereby φ denotes the angle between the y-axis and the vetor x S x 0. With osφ = equation (33) beomes y S x S x 0 j ω D trad (x 0, ω) = Ŝ(ω) 2π x S x 0 x ref x 0 x S x 0 + x ref x 0 y S x S x 0 (34) e j ω xs x0. x S x 0 (35) For x S x 0, thus when the virtual soure is positioned far behind the seondary soures, (35) simplifies to D trad (x 0, ω) Ŝ(ω) j ω 2π x ref x 0 y S x S x 0 e j ω xs x0 x S x 0. (36) The proposed driving funtion D sw,2.5d (x 0, ω) for a linear seondary soure distribution reproduing a monopole point soure at position x S inluding 2.5D orretion an be found in equation (29). In the following we assume that the seondary soure distribution is positioned parallel to the x-axis at y = y 0. For x S x 0 the addend / ω x S x 0 in (29) beomes insignifiant ompared to the added jω/ and (29) then reads D sw,2.5d (x 0, ω) Ŝ(ω) 2πj ω y S x ref x 0 x S x 0 e j ω xs x0 x S x 0. (37) Equation (36) and (37) are similar exept for a normalization fator. Thus, when the virtual sound soure is suffiiently far behind the seondary soure distribution ( x S x 0 ), the two driving funtions are equal. 5. ARTIFACTS OF WFS The following setion will briefly disuss various artifats that emerge from pratial aspets and the assumptions made to derive the driving funtions. Note that most of these artifats may also be present in other sound field reprodution tehniques, like for instane higher-order Ambisonis. 5.. Spatial Sampling of Seondary Soure Distribution The theory presented so far assumes a spatially ontinuous distribution of seondary soures. Pratial implementations of WFS will onsist of seondary soures that are plaed at spatially disrete positions. This spatial sampling of the ontinuous distribution may lead to spatial aliasing artifats in the reprodued wave field. Spatial aliasing onstitutes a disturbane of the spatial struture of the reprodued wave field. Therefore, it potentially may result in loalization inauraies and oloration artifats. The effets of spatial sampling on the reprodued wave field have been evaluated in the literature on WFS [42, 44, 2, 45]. An analysis of aliasing artifats is only possible when onsidering a partiular geometry of the seondary soure distribution ontour. However, two main onlusions an be drawn: () spatial aliasing inreases with the bandwidth Page 3 of 9

14 of the virtual soure signal and (2) spatial aliasing artifats depend on the listener position. Typial WFS systems employ loudspeakers with a spaing of x = m. The resulting spatial aliasing artifats beome prominent for frequenies of the virtual soure of roughly above khz (spatial aliasing frequeny). This would indiate that WFS annot be used for the auralization of typial audio soures. However, the human auditory system seems to be not too sensible to spatial aliasing if the loudspeaker spaing is hosen in the range x = m. A detailed analysis of the pereptual impat of spatial aliasing is an urrent researh topi. The results in [] indiate that spatial aliasing may lead to oloration. The various WFS driving funtions derived in this paper ontain a pre-filtering of the virtual soure signal with j ω or j ω harateristi, respetively. The pre-filtering of the virtual soure signal should only be performed below the spatial aliasing frequeny, sine the theoretial foundation for the pre-filtering holds only there. A flat response above the spatial aliasing frequeny has proven to be suitable in pratie. Sine the spatial aliasing frequeny is different for different listener positions this may lead to additional oloration artifats for listener positions the filter hasn t been optimized for Trunation of Seondary Soure Distribution Pratial implementations of seondary soure distributions with non-losed ontours will always be of finite size. The theory of WFS assumes losed ontours, infinitely long linear or infinitely sized planar distributions. The trunation of the seondary soure distribution leads to artifats. These artifats in the reprodued wave field are referred to as trunation artifats in the ontext of WFS. The effets emerging from the trunation of linear seondary soure ontours used for WFS have been investigated in detail by [7, 6, 46]. The effet of trunating the length of a linear seondary soure distribution an be qualitatively understood as the effet a gap has on a propagating wave field. Two effets an be observed [46]: () the area of the orretly reprodued wave field is limited by the finite aperture and (2) irular waves propagate from the outer seondary soures. The first effet an be desribed by ray theory, the latter by diffration theory. Note, that due to the separability of the Cartesian oordinate systems these onsiderations also hold for planar seondary soure distributions. It has been shown that trunation artifats an be redued by applying a weight (tapering window) to the seondary soure driving signals. Typially a one-sided squared osine window is applied to the loudspeaker driving signals in order to redue the artifats. As a side effet, tapering will redue the effetive listening area. Depending on the desired virtual soure field, the seondary soure seletion riterion (8) may limit the number of ative loudspeakers. This onstitutes essentially a trunation of the seondary soure ontour and may lead to trunation artifats. It was proposed in [3] to apply a tapering window to the driving signals whih depends on the ative seondary soures and the atual virtual soure to be reprodued Amplitude Errors Two-dimensional WFS systems typially use point soures (or their approximations by loudspeakers) as seondary soures. As already outlined in Setion 4.2, this seondary soure type mismath leads to amplitude errors in the reprodued wave field. An detailed analysis of these amplitude errors an be found in [47, 6]. For the reprodution of virtual plane waves, these amplitude errors have the onsequene that the reprodued wave field will exhibit an amplitude deay within the listening area. The resulting amplitude deay is approximately 3 db per doubling of distane. The reprodued wave field for a virtual spherial wave will exhibit an amplitude deay whih is inbetween the amplitude deay of a ylindrial wave and a spherial wave Other Artifats Two other artifats of WFS are disussed briefly in the following. The first artifat is related to the reprodution of moving virtual soures, the other to the reprodution in a plane only. Moving virtual point soures are typially reprodued by using the model of a stationary spherial wave and hanging its position over time. This leads to various artifats as reported in [37, 38]. It was proposed in this ontext to use the model of a moving point soure to derive the seondary soure driving funtion. This way some of these artifats are resolved. However, the results presented in [38], Page 4 of 9

15 using suh a model, indiate that spatial aliasing and trunation artifats play a more prominent role in the auralization of moving virtual soures than for stationary soures. The reprodution in a plane only (two-dimensional WFS) using point soures as seondary soures will lead to artifats for listeners whih are not loated in the plane where the loudspeakers are. Its not always possible to level the loudspeakers with the listeners ears due to tehnial restritions. As a onsequene out of plane listeners will have the impression that the virtual soures are elevated or lowered. 6. EXTENSIONS TO WFS So far, the basi theory of WFS and some of its artifats have been disussed. However, WFS has been improved in various diretions to ope with the problem of spatial aliasing, the listening room aoustis, the properties of real loudspeakers and noise soures. These extensions are briefly reviewed. 6.. Enhanement of Spatial Aliasing The relatively low spatial aliasing frequeny of typial WFS systems implies potential problems in terms of oloration and loalization of virtual soures. Various tehniques have been proposed to enhane the situation. It is proposed in [48, ] to ombine WFS with stereophoni tehniques in order to improve oloration artifats that arise from spatial aliasing. This tehnique has been termed as optimized phantom soure imaging (OPSI). The WFS driving funtions are used below the spatial aliasing frequeny, while above the spatial aliasing frequeny amplitude panning with seleted loudspeakers is used. The results reported so far show that OPSI provides the potential to redue oloration artifats while preserving most of the good loalization properties of WFS. Another tehnique proposed to improve the pereption of spatial aliasing artifats is to randomize the phase above the spatial aliasing frequeny in the driving funtion [5]. The basi idea of this approah is to smear out the spatial struture of spatial aliasing. The results reported in [49] using diffuse filters show some potential Ative Listening Room and Loudspeaker Compensation The basi theory of WFS, as presented in Setion 2, relies on free-field wave propagation and does not onsider the influene of refletions within the listening environment. Sine these refletions may impair the arefully designed spatial sound field, their influene should be minimized by taking appropriate ountermeasures. Besides passively damping of the listening room various ative tehniques have been proposed e. g. [50, 5, 52, 53, 54, 55, 47, 0, 56, 57, 58, 59, 60, 6]. Common to all of these approahes is that they perform a pre-filtering of the loudspeaker driving signals in order to anel the listening room refletions by destrutive interferene. However, suitable ontrol is only ahievable when no spatial aliasing is present in the reprodued wave field. Due to the varying harateristis of the propagation medium and the aousti environment an adaptive omputation of these pre-filters on basis of the reprodued wave field is desirable. Simulation results indiate that ative listening room ompensation provides the potential to redue the effets of the listening room. Besides the influene of the listening room, also nonideal harateristis of the seondary soures may degrade the reprodued wave field. The ompensation of non-ideal seondary soure harateristis is known as loudspeaker ompensation in the ontext of WFS. Loudspeaker ompensation an be seen as subset of listening room ompensation. However, no adaptive filters are required in this ase sine the harateristis of the loudspeakers an be assumed to be onstant over time. Multihannel tehniques have been proposed to perform loudspeaker ompensation [62, 2] Ative Noise Control Another kind of impairments are noise soures within the listening room. If these soures are loated outside the listening area, ative noise ontrol (ANC) tehniques an be applied. The signal proessing tehniques used for ANC are similar to those used for ative listening room ompensation. Hene, the same algorithms an be applied in priniple to ANC for WFS. First results an be found in [63, 64, 65, 66] 7. CONCLUSION This paper presents a generalized theory for WFS. Three-dimensional reprodution in a listening volume, as well as two-dimensional reprodution in a listening area is overed. In the latter ase, both the Page 5 of 9