Page  00000001 ELEVATION PROBLEMS IN THE AURALISATION OF SOUND SOURCES WITH ARBITRARY SHAPE WITH WAVE FIELD SYNTHESIS M.A.J. Baalman Technische Universitat Berlin Department of Communication Sciences baalman@kgw. tu-berlin. de ABSTRACT In order to reproduce arbitrarily shaped sources with Wave Field Synthesis, source points outside of the horizontal plane have to be taken into account. The loudspeaker driver function for these points is derived mathematically. The errors made for elevated points are discussed and simulations show the effect of these errors. 1. INTRODUCTION Wave Field Synthesis (WFS) offers many possibilities for spatialisation of electronic music. Its main advantage is that it has no sweet spot, but instead a large listening area, making the technology attractive for concert situations. Yet, whenever acoustic instruments are combined with electroacoustic sounds in concerts, is that the electroacoustic part tends to lack depth and extension in comparison with the sound from the acoustic instruments. Misdariis [1] and Warusfel [2] have proposed special 3D loudspeaker arrays to simulate different radiation characteristics. A problem with this technique is that the object itself is static; a movement of the source can only be created by physically moving the loudspeaker array. In current Wave Field Synthesis applications there are only solutions for monopole point sources and plane waves. There are some reports of ad hoc solutions to simulate larger sources, such as the Virtual Panning Spots [3] and the auralisation of a grand piano [4] by using a few point sources. Currently, work is done on implementing radiation characteristics for point sources [5]. By definition radiation characteristics are only applicable at a certain distance from the object modelled. Since WFS sources can get very close, it makes sense to do research to find a general solution for auralising an arbitrarily shaped source object. In this paper the derivation of the 2 D-operator is reviewed for the case of points outside of the horizontal plane, the errors that occur by using this operator are discussed and finally these are investigated in simulations. 2. THEORY For WFS reproduction of a 3-dimensional source object the commonly used 2 D-operator [6] is not sufficient as in the derivation only sources in the same plane as the array and reference line are taken into account. Thus, in order to derive a proper driver function, we must start anew from the Rayleigh integrals. The Rayleigh I integral states that if we know the normal component (pointing towards receiver area) of the velocity V, on a plane S caused by a source distribution on the left hand of S, we can determine the pressure in point R in the receiver plane by 1 j, e-jkAr P(r) = j wpoV ( ) dS (1) in which po denotes the medium density and Ar Ar | rI - rS, i.e. the distance from the receiver point rR to a point rs on the surface S (see figure 1). k is the wavenumber. 4i r ^r 6:1. 2 *- Ia 'v ^ s. i ~\/ 1' -7- ^ I ' R, Figure 1. Geometry for derivation of the 2 D-operator. T1 and I2 are points from the source distribution, r' and ri the vectors to a point M on the integration line m. Ar is the vector from a point M on the integration line to the receiver point R on the line 1. A sound source of arbitrary shape can be modelled by regarding the source as a collection of point sources; the sum of their wavefields is the resulting wave field. Hence, substituting the far field approximation (kr > 1) of the normal component of the particle velocity on the plane S due to a monopole sound source into the Rayleigh I synthesis integral yields

Page  00000002 w P(reo) 1 Pm(seiu)d with Pm the pressure on the line m (see figure 1): (2) Prn(Tj) IJkS(W)cos e-jkr C-jkAr T'o Q 0 Ar dy (3) z We can now apply the stationary phase method ([7], ch. 2.7) to equation 3, as the integrand is of the form: I = f (y)e (Y)dy (4) with V)(y) -k(r + Ar). Now we need to find the stationary point of V)(y). Then, for | ~> 1, the approximate solution to 4 is I =f (yo)eie~vo 2x) j) " ( W o)j f (YOC (Y(5) where yo is the stationary phase point'1 and V"(yo) is the second derivative of V)evaluated at yo. From inspection of figure 2, it follows that if R is any point on line 1, the point yo should lie in the plane through T and I and thus: Yo= YR + (YwI- YR) ZR (6) ze + zR and at the stationary point: (Yo) = -k(ro + Aro), (7) ro + A To V)"(Yo) -kTo+Arro (8) f(YO) 1 S(w)( 27TroArocos~o (9) where zR is the z-coordinate of the receiver and zp of the source point. The driver function for a speaker for the contribution of one monopole source point then becomes: Q(x, w) = jk Aro e-jkro S(w) cos(Ao) (10) 27 Aro + 0o r0 where x is the coordinate of the speaker at the speaker array (assumed to be at the x-axis). It should be noted that the actual elevation will not be heard by the receiver, when the elevated source is played back by the WFS-array. The elevated points are mainly of interest, because their contributions will interfere with those of the points in the horizontal plane. 1 for each line m (x = xm and z = 0) there is a stationary point with the y-coordinate yo. X Figure 2. The stationary point yo lies on the cross-section of plane S and the plane through T and R. In practice Aro will in fact be Aro. 3. ERRORS INTRODUCED FOR ELEVATED SOURCE POINTS For point sources outside of the horizontal plane yo # 0. This means that the main contribution of this point should not be emitted from the speaker position, but from a higher or lower position. A small error will be made as the actual propagation of the source signal will be from the speaker to the receiver point and not from the stationary point to the receiver point (see figure 2). This means that the amplitude will be higher and the delay will be shorter. Note that the error will be larger for sources further away from the horizontal plane and for sources closer to the array (for constant yp). This error can be compensated by adding an extra factor of Aro e-jk(Aro-Aro) Ar0 (11) to the driver function (eq. 10). In figure 3 the errors in the delay and amplitude are shown for the source-speaker-listener path. The reference is the direct path from source to receiver. The errors are shown for the contribution of 1 speaker of the array, that is directly in between the source and receiver (i.e. the xcoordinates of source, speaker and receiver point are the same). The receiver point is at the reference line. The figure shows that the compensation factor corrects the error. 3.1. Errors for receiver points not on the reference line The derivation of the driver function (10) is valid for a reference line. For receiver points not on this line errors will be made that may further influence the interference pattern of the waves coming from elevated points2 For points closer to the array than the reference line, the stationary point yo would be smaller than the one used 2 Sonke ([8], ch. 4.3) discussed the amplitude error for receiver points not on the reference line made for points in the horizontal plane

Page  00000003 ys = 0.5 ',li " - ~-... 2 8 1 re eier(m Scompensated ____________ ______________________ Figure 3. The delay and amplitude error for source points at different elevations and three different distances behind the array. Note that the correction factor eliminates the error. Figure 5. The error in delay and amplitude for receiver points not on the reference line, which is at 3m from the array. (11). Yo,f I I YO,b _ Ar0 -5 m *k,; I I Ry rR, rRb y -3 m 1 -1 m speaker array i>t.I1.m' S. ''- 3 m = reference line S" X,5.mn z Figure 4. Geometrical explanation of the error made for receiver points not on the reference line. For receiver points Rf in front of the reference line R the stationary point yo,f would be lower than yo. For receiver points Rb behind the reference line R the stationary point YO,b would be higher than yo. for the reference line (see figure 4). Hence, ro should be larger for these receiver points and thus an error is made in delay and amplitude of the driver function. Aro will be smaller for these points, resulting in further errors in the propagation delay and in the amplitude factor. For points that are further from the array than the reference line, the effects are opposite. In figure 5 the errors are visualised for the contribution of one speaker. The figure shows the difference in delay and amplitude, when sound is propagated from source to receiver position. The reference line is taken at 3 meters from the array. The error is up to 20 ms in delay time and up to 5 dB in amplitude. This is an order of magnitude higher than the error described in the previous subsection. The effect of an error in the delay will be that the phases of the waves arriving at the listening point will not be correct and the interference between wave fields of different source points will be distorted; in the worst case certain frequency components may be amplified instead of attenuated through interference, causing a false frequency spectrum for the listener. The amplitude effects also influence the interference pattern. 4. SIMULATIONS 4.1. Setup A number of point sources at different elevations, running from -0.5 to +0.5 m in steps of 0.1m, centered behind the WFS array, was used as the source for simulations done x Figure 6. Geometry of the simulations. Source points are sampled at 0.1m intervals in y-direction. The speaker array and the receiver lines are sampled at 5cm intervals. The speaker array goes from -3 to 3m, the receiver lines from -3.5 to 3.5m in x-direction. with Matlab. The source signal was a gaussian wavelet with no frequency components above 750Hz. The source signal was extrapolated with a wave field extrapolation operator based on the Rayleigh I integral. In both simulations, the following source and receiver distances were used: 1, 3 and 5 meter behind the array and 1, 3 and 5 meters in front of the array (see figure 6). The reference line was at 3 meters in front of the array, parallel to it. In order to avoid truncation effects, a Hanning window over 20% of the array length was used on both sides (see [6]). Direct extrapolation from source to listening position was compared with WFS reproduction followed by extrapolation from array to listening position. The WFS reproduction array was sampled at 5cm intervals, as were the listening positions. The corresponding aliasing frequency for this array is 3.4kHz, thus we can safely assume that the WFS reproduction does not introduce aliasing problems. 4.2. Results In figure 7 the interference pattern is shown of a source in the horizontal plane and a source at 0.lm height at 3m behind the array, calculated by adding up the extrapolated wave fields. Both WFS reproductions show spatial irregularities: in the frequency spectrum there are dips on the top of frequency maxima depending on receiver location

Page  00000004 Figure 7. Interference pattern of two point sources at Om and 0.1m elevation. Lefthand: direct wave field extrapolation (wfe), middle: WFS reproduction with driver function (10) and righthand: WFS reproduction with the compensated driver function ((10) with (11)). and the spectrum is not as constant over the spatial area as it should be. In figure 8 the interference pattern between five points with elevations from 0 to 0.5m with 0.1m distance are shown. Here the WFS reproduction is even more irregular. In the time domain, we see a wavefront with small amplitude ahead of the actual wavefront in the WFS reproduction. This preliminary wavefront is stronger with the compensated driver function. The compensation factor makes a correction for the delay and amplitude with respect to the difference between speaker position and stationary phase point; however, it seems that when this factor is included in the driver function, the interference takes place too early and undesired artefacts are created. It will be up to subjective evaluation whether or not the compensation factor should be applied. 5. CONCLUSIONS AND FUTURE WORK A derivation of the 2 D-operator for source distributions was made. It was shown that errors will be made for source points outside the horizontal plane and a compensation factor was proposed for the propagation error. The errors in amplitude and delay were verified in simulations: the frequency spectrum is distorted at some points in the listening area due to the interference of elevated points that are played back over WFS. Listening experiments will have to evaluate the audibility of the errors. An implementation will be done in the software WONDER [9]. 6. REFERENCES [1] N. Misdariis, O. Warusfel & R. Causse. Radiation control on a multi-loudspeaker device. ISMA 2001, 2001. [2] N. Misdariis & T. Caulkins O. Warusfel, E. Corteel. Reproduction of sound source directivity for future audio applications. ICA 2004, 2004. Figure 8. Interference pattern of five point sources between Om and 0.5m elevation at 0.1m distance from each other in the time (top) and frequency (bottom) domain. Plots from left to right as in figure 7. [3] G. Theile, H. Wittek & M. Reisinger. Wellenfeldsynthese Verfahren: Ein Weg fiir neue M6glichkeiten der riumlichen Tongestaltung. 22nd Tonmeistertagung, Hannover, Germany, 2002 November, 2002. [4] I. Bork, & M. Kern. Simulation der Schallabstrahlung eines Fluigels. DAGA '03, 2003. [5] 0. Warusfel & N. Misdariis. Sound source radiation synthesis: from stage performance to domestic rendering. AES 116th Convention, Berlin, Germany, 2004, May, Preprint 6018, 2004. [6] E.N.G. Verheijen. Sound Reproduction by Wave Field Synthesis. PhD thesis, TU Delft, The Netherlands, 1998. [7] N. Bleistein. Mathematical Methods for Wave Phenomena. Academic Press, New York, 1984. [8] J.J. Sonke. Variable Acoustics by Wave Field Synthesis. PhD thesis, University of Technology, Delft, 2000. [9] M.A.J. Baalman & D. Plewe. Wonder - a software interface for the application of wave field synthesis in electronic music and interactive sound installations. International Computer Music Conference 2004, Miami, 1-6 November 2004, 2004. 2004.