A preliminary investigation is presented of horizontal sound field control using a rectangular array of spherical wave driving sources, with focus on wide area plane wave reproduction. Comparison is made between Wave Field Synthesis and a method based on distributed pressure and velocity constraints. Advantages of the second approach are discussed, and modifications of the both methods are propose

Menzies, Dylan

Southampton (e-Prints Soton)

Horizontal Sound Field Reproduction with Point Sources usingDistributed ConstraintsD. MenziesDe Montfort University, Email: rdmg@dmu.ac.ukAbstractA preliminary investigation is presented of horizontal sound field control using a rectangular array of spherical wavedriving sources, with focus on wide area plane wave reproduction. Comparison is made between Wave Field Synthesisand a method based on distributed pressure and velocity constraints. Advantages of the second approach are discussed,and modifications of the both methods are proposed.IntroductionHorizontal sound field reproduction with point drivingsources using either Wave Field Synthesis (WFS) [1, 2, 3]or High Order Ambisonics (HOA) [4, 5] suffers fromreproduction errors that do not fall to zero as the numberof drivers is increased. The error is unsurprising becausea 3D source restricted to the plane is not a solution ofthe 2D wave equation, most obviously because the 2Dfield divergence is not zero around the 3D source. A linearray with 3D drivers can be analyzed fully because ofits symmetry, however it is not immediately clear what ispossible with more general arrays. In an attempt to findbetter solutions a distributed constraint (DC) method isinvestigated here following on from related work using 2Dand 3D drivers for 2D and 3D control. Simulations areused to compare the fields generated using WFS and DC.The test fields used are plane waves, which are harderto reproduce accurately than virtual point source fieldsdue to the use of more drivers. The arrays used arerectangular which is a common practical configuration,particularly in concert halls. From the scattering view inthe Simple Source formulation [6], driving functions areexpected to be more complex for boundaries with sharpercorners.Reproduction methodsDistributed constraintsPreviously the control of 2D sound fields was consideredby applying modal constraints on a grid of points inthe target region [7]. This is effective because theconstraints at each point can accurately describe a soundfield region of any radius. In 3D with a horizontal arraythe contribution from each driver at a point in the arrayplane cannot be expressed in a 2D modal expansion.Of course it is possible to apply derivative constraintsof any order, but the non-attainability of the solutionimplies that higher-order constraints will dominate theerror and the problem will be over-constrained. For thisreason only pressure and gradient constraints are givenat each point in the target region. This is equivalent toconstraint by the horizontal components of the first orderFourier-Bessel expansion.The set of all constraints can be written at each frequencyascln =∑Sjn(rsj − rcl)sj =∑Mlnjsj . (1)where cln are the pressure and horizontal gradient,indexed by n at each constraint point, indexed by l. sjare the driving functions to be determined. Sjn(rsj −rc)are the pressure and horizontal gradient of the jth driverfield located at rsj − rc relative to the measurementpoint. rsj and rc are the locations of the sources and theconstraints respectively. In the case of monopole driversassumed in this study, the driver field descriptions areidentical, Sjn = Sn.The driving functions are found by finding the pseudoinverse of Mlnj with l and n grouped into one index,s = M¯+c¯ . (2)Basic Tikhonov regularization is used, so that M+ isthe regularized pseudo inverse (MTM+α2I)−1MT , withparameter α.The driver normalization used is eikr/4πr, the same asthat matching (4). This allows the driving function levelsto be compared directly. The gradient of the driving fieldin the horizontal plane is equal to the total gradient,∇e−ikr4πr= −(ik +1r)e−ikr4πrrˆ . (3)This is equivalent to the horizontal components in thefirst order Fourier-Bessel expansion of the field. havingthe same r dependence given by the spherical hankelfunction, h(2)1 (kr).Wave Field SynthesisWFS is a well established method for efficiently calcu-lating and applying driving functions for the synthesis ofvirtual sources. For the synthesis of a plane wave e−ik·ron a horizontal array the continuous driving function is[3]Dpw,2.5(x0, ω) = −2apw(x0)√2π|xref − x0| ××√iwcSˆpw(ω)nTpwn(x0)e−i ωcnTpw .x0 (4)This is sampled at each driver point with an extrafactor δs, the driver spacing, included to match thelevel of the continuous driver reproduction. The drivingfunction consists of a driver independent filter, and driverdependent gain and delay terms.SimulationsFor convenience length units are chosen so that thehighest frequency considered, the WFS alias / Nyquistfrequency, has wavelength λ = 1. The array spacing isthen λ = 0.5. The test array used has dimensions 10×10.The length in metres of the array spacing used in WFSsystems is typically 0.10m to 0.3m corresponding to aNyquist frequency of ≈ 1000Hz. The real arrays withthis spacing corresponding to the test arrays would havedimensions ≈ 4m × 4m. Although concert halls can beseveral times larger than this, the results will provide auseful guide.The first set of simulations show the reproduction of aplane wave traveling at two angles, 0◦, 45◦ relative to theright pointing direction. These are repeated for WFS andDC, and for λ = 1, 4, Figures 1,2,3,4,5,6,7,8. For DC theconstraints are placed on a square grid with spacing 0.25. The grid is separated from the boundary by a marginof distance 1. For each simulation the following plotsare shown : the real value of the reproduced complexpressure field P (x, ω); the relative error ǫ(x, ω) =|P (x, ω)−Pt(x, ω)|/Pt(x, ω) where Pt(x, ω) is the target;and the absolute value |P (x, ω)| expressed in dB. Thetarget planewave is 0 dB everywhere. Also shown on thetop line of each figure is max(s), the maximum absolutevalue of the driving function, as an indication of driverenergy level.The DC simulations show good coverage and wavefrontstability, and level fade in the direction of travel thatis significantly reduced relative to WFS. Directionalimaging stability is indicated by the variation of thewave gradient direction, which is clearly visible. TheWFS simulations exhibit the level fade known from linearrays, and also a narrowing of width. The small scaleinstability can be nearly removed at λ = 1 by modifyingthe selection window so that it fades over distance 2 ateach edge. At lower frequencies this approach is notcompletely effective, and instability remains, as wouldbe expected from the Simple Source scattering picture.The narrowing effect worsens progressively as edge fadeis introduced.The next set of simulations shown in Figures 9,10,11,12model a reduced target area, starting distance 4 from theleft, which could represent a concert room with a stagearea on the left where a coherent field is not required.The increased freedom permits solutions with furtherRe(P(x,ω))λ=1.0, δs=0.5, max(s)=13.202468100 2 4 6 8 10-2-1.5-1-0.500.511.52ε(x,ω)02468100 2 4 6 8 1000.10.20.30.40.520 log(|P(x,ω)|)   (dB)02468100 2 4 6 8 10-6-4-20246Figure 1: WFS reproduction, plane wave angle 0◦, λ = 1,reduced fade and stable wavefronts, perhaps more thanmight be expected. Evidently the increased frontaldistance working in combination with side sources iseffective. Solutions for greater angles show more similarresults to those found in the previous set, as would beexpected.WFS modificationIt appears the availability of side drivers in DC is effectiveat compensating the narrowing effect seen in WFS,particularly at small angles. This suggests WFS mightbe modified by extending the driving function so it isactive along the sides in order to reduce narrowing. Afirst attempt is given by the following modification of thedot product factor,nTpwn(x0)→ (nTpwn(x0) + γ)/(1 + γ) . (5)The driving function is unchanged where the boundaryfaces in the direction of the plane wave. Simulationsusing this factor are shown in Figures 13,14Re(P(x,ω))λ=1.0, δs=0.5, δc=0.25, α=0.1, max(s)=11.402468100 2 4 6 8 10-2-1.5-1-0.500.511.52ε(x,ω)02468100 2 4 6 8 1000.10.20.30.40.520 log(|P(x,ω)|)   (dB)02468100 2 4 6 8 10-6-4-20246Figure 2: DC reproduction, plane wave angle 0◦λ = 1,Narrowing and fade is reduced at both frequencies shown.Wavefront stability is not very good but improves atthe lower frequency. It appears that an approximateevanescent field produced by the sides is partly com-pensating for narrowing effect. This positive resultraises the question of treating the main components ofthe WFS driving function, pre-delay, gain and delayas free parameters in order to optimize reproductionefficiently. The optimal control of the parameters mightbe encoded using efficient non-linear learning schemes,such as gaussian mixtures or neural networks. Thecontrol of Further more the driving function mightbe given more degrees of freedom, while remainingcomputationally efficient, for example by introducing anadditional pre-filter:Dl = (f1(ω)g1,l + f2(ω)g2,l)e−iwtl (6)Re(P(x,ω))λ=4.0, δs=0.5, max(s)=6.602468100 2 4 6 8 10-2-1.5-1-0.500.511.52ε(x,ω)02468100 2 4 6 8 1000.10.20.30.40.520 log(|P(x,ω)|)   (dB)02468100 2 4 6 8 10-6-4-20246Figure 3: WFS reproduction, plane wave angle 0◦λ = 4,ConclusionThe simulations have shown that at least for plane wavereproduction in rectangular boundaries DC can improveon the reproduction quality of WFS, in terms of stability,width narrowing and distance fade, over 3 octaves fromthe WFS alias frequency (not all shown). This suggestsgood broadband performance, however more simulationsare needed to test pulse performance. Reduction of targetarea, for instance due to a stage area can give significantimprovements over a range of angles.The cost of both filter calculation and filter applicationis much higher in DC than in WFS, requiring FIRfiltering. However it could be useful to pre-calculatefilters for static sources. Furthermore, the playback ofcomplete pre-rendered driving signal sets is possible fromconsumer computer equipment together with audio in-terfaces already in use with WFS systems. For sufficientrendering complexity it may eventually make sense topre-render filter sets for basis fields covering the wholetarget area. Different methods can be combined, so forexample combining near WFS sources with pre-renderedRe(P(x,ω))λ=4.0, δs=0.5, δc=0.25, α=0.1, max(s)=5.902468100 2 4 6 8 10-2-1.5-1-0.500.511.52ε(x,ω)02468100 2 4 6 8 1000.10.20.30.40.520 log(|P(x,ω)|)   (dB)02468100 2 4 6 8 10-6-4-20246Figure 4: DC reproduction, plane wave angle 0◦λ = 4,distant backgrounds.Other areas for consideration are the modelling of speakerdirectivity to improve reproduction accuracy. Pre-fadingthe target solutions to more closely match what ispossible may help to produce better solutions. Alsoit was shown that WFS itself can be modified withoutsignificant extra cost to give improved reproduction.There are many ways in which this could be furtherexplored.References[1] A. J. Berkhout and P. de Vries, D.and Vogel,“Acoustic control by wave field synthesis,” J. Acoust.Soc. Am., vol. 93, pp. 2764–2778, 1993.[2] E. Verheijen, “Sound reproduction by wave fieldsynthesis,” Ph.D. dissertation, Delft University ofTechnology, January 1998.[3] S. Spors, R. Rabenstein, and J. Ahrens, “The theoryof wave field synthesis revisited,” in Preprint 7358,AES 124th Convention, Amsterdam, May 2008.Re(P(x,ω))λ=1.0, δs=0.5, max(s)=9.302468100 2 4 6 8 10-2-1.5-1-0.500.511.52ε(x,ω)02468100 2 4 6 8 1000.10.20.30.40.520 log(|P(x,ω)|)   (dB)02468100 2 4 6 8 10-6-4-20246Figure 5: WFS reproduction, plane wave angle 45◦, λ = 1,[4] M. A. Gerzon, “Periphony (with-height soundreproduction),” in Proc. AES 2nd Convention,Munich, 1972.[5] J. Daniel, “Spatial sound encoding including nearfield effect,” in Proc. AES 23nd InternationalConference, Helsinger, Denmark, 2003.[6] E. Williams, Fourier Acoustics: sound radiation andnearfield acoustical holography. Elsevier, 1999.[7] D. Menzies, “Sound synthesis for general enclosures,”in Proc. 2nd Int. Symposium on Ambisonics andSpherical Acoustics, IRCAM Paris, May 2010.Re(P(x,ω))λ=1.0, δs=0.5, δc=0.25, α=0.1, max(s)=11.202468100 2 4 6 8 10-2-1.5-1-0.500.511.52ε(x,ω)02468100 2 4 6 8 1000.10.20.30.40.520 log(|P(x,ω)|)   (dB)02468100 2 4 6 8 10-6-4-20246Figure 6: DC reproduction, plane wave angle 45◦λ = 1,Re(P(x,ω))λ=4.0, δs=0.5, max(s)=4.702468100 2 4 6 8 10-2-1.5-1-0.500.511.52ε(x,ω)02468100 2 4 6 8 1000.10.20.30.40.520 log(|P(x,ω)|)   (dB)02468100 2 4 6 8 10-6-4-20246Figure 7: WFS reproduction, plane wave angle 45◦λ = 4,Re(P(x,ω))λ=4.0, δs=0.5, δc=0.25, α=0.1, max(s)=13.702468100 2 4 6 8 10-2-1.5-1-0.500.511.52ε(x,ω)02468100 2 4 6 8 1000.10.20.30.40.520 log(|P(x,ω)|)   (dB)02468100 2 4 6 8 10-6-4-20246Figure 8: DC reproduction, plane wave angle 45◦λ = 4,Re(P(x,ω))λ=1.0, δs=0.5, δc=0.25, α=0.1, max(s)=16.002468100 2 4 6 8 10-2-1.5-1-0.500.511.52ε(x,ω)02468100 2 4 6 8 1000.10.20.30.40.520 log(|P(x,ω)|)   (dB)02468100 2 4 6 8 10-6-4-20246Figure 9: DC reproduction, reduced target, plane waveangle 0◦λ = 1,Re(P(x,ω))λ=4.0, δs=0.5, δc=0.25, α=0.1, max(s)=8.402468100 2 4 6 8 10-2-1.5-1-0.500.511.52ε(x,ω)02468100 2 4 6 8 1000.10.20.30.40.520 log(|P(x,ω)|)   (dB)02468100 2 4 6 8 10-6-4-20246Figure 10: DC reproduction, reduced target, plane waveangle 0◦λ = 4,Re(P(x,ω))λ=1.0, δs=0.5, δc=0.25, α=0.1, max(s)=13.102468100 2 4 6 8 10-2-1.5-1-0.500.511.52ε(x,ω)02468100 2 4 6 8 1000.10.20.30.40.520 log(|P(x,ω)|)   (dB)02468100 2 4 6 8 10-6-4-20246Figure 11: DC reproduction, reduced target, plane waveangle 45◦λ = 1,Re(P(x,ω))λ=4.0, δs=0.5, δc=0.25, α=0.1, max(s)=8.802468100 2 4 6 8 10-2-1.5-1-0.500.511.52ε(x,ω)02468100 2 4 6 8 1000.10.20.30.40.520 log(|P(x,ω)|)   (dB)02468100 2 4 6 8 10-6-4-20246Figure 12: DC reproduction, reduced target, plane waveangle 45◦λ = 4,Re(P(x,ω))λ=1.0, δs=0.5, max(s)=13.202468100 2 4 6 8 10-2-1.5-1-0.500.511.52ε(x,ω)02468100 2 4 6 8 1000.10.20.30.40.520 log(|P(x,ω)|)   (dB)02468100 2 4 6 8 10-6-4-20246Figure 13: Modified WFS reproduction, γ = 0.2, plane waveangle 0◦λ = 1,Re(P(x,ω))λ=4.0, δs=0.5, max(s)=6.602468100 2 4 6 8 10-2-1.5-1-0.500.511.52ε(x,ω)02468100 2 4 6 8 1000.10.20.30.40.520 log(|P(x,ω)|)   (dB)02468100 2 4 6 8 10-6-4-20246Figure 14: Modified WFS reproduction, γ = 0.2, plane waveangle 0◦λ = 4,

Horizontal sound field reproduction with point sources using distributed constraints

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