DSP Formulation of a Finite Difference Method for Room Acoustics ...

600AxialGroup delays (in samples)40020000 0.05 0.1 0.15 0.2 0.256004002002−D Diagonal00 0.05 0.1 0.15 0.2 0.256004002003−D Diagonal00 0.05 0.1 0.15 0.2 0.25Normalized frequency (Nyquist == 0.5)Fig 6. Group delays in three different directions, axial,2D diagonal and 3D diagonal.( 1 + r) ⋅ z – 1 poppp = -----------------------------------------(11)1 + rz – 2This result is found by using equation (4) and setting+p wall= 0 , since there is no input from inside the wall.In addition the following relation between impedanceZchange and reflection factor is used: r 2 – Z= ------------------ 1. Equation(11) can also be expressed as a difference equationZ 2 + Z 1pn ( ) = ( 1 + r) ⋅ p opp( n – 1)– r ⋅ p( n–2)(12)where p is the boundary node and p opp is its neighbor. Ingeneral the boundary condition can be replaced by anydigital filter.Figure 7 shows behaviour of some simple boundaryconditions. The model consists of three rooms constructedof different materials. The first figure shows the roomconfiguration, excitation shape and first order phase preservingreflection from the ceiling. In the second one thereare phase reversing reflections from the walls. The wallsseparating rooms are anechoic.5. SUMMARY AND FUTURE WORKIn this paper we have shown that finite difference time domainmethod can also be formulated in terms of digitalsignal processing. The study is based on digitalwaveguides.In the future we are going to make this algorithm tobetter suit real room acoustic problems. This includes adetailed study on boundary conditions and also on couplingof different sound propagation mediums. To enhancedispersion characteristics various interpolationtechniques are under study, and some results are describedin a companion paper [9].Fig 7. Visualization of simulation, excitation andfirst order reflection (above), wall reflections (below).REFERENCES[1] D. Botteldooren, “Finite-difference time domain simulationof low-frequency room acoustic problems,” J.Acoust. Soc. Am., vol. 98, no. 6, pp. 3392-3308, 1995.[2] J. Smith, “Physical modeling using digitalwaveguides,” Computer Music Journal, vol. 16, no.4, pp. 75-87, Winter 1992.[3] S. Van Duyne and J. Smith, “Physical modeling withthe 2-D digital waveguide mesh,” In Proc. 1993 Int.Computer Music Conf., Tokyo, Sept. 10-15, 1993, pp.40-47.[4] L. Savioja, J. Backman, A. Järvinen, and T. Takala,“Waveguide mesh method for low-frequency simulationof room acoustics,” In Proc. 15th Int. Congresson Acoustics (ICA’95), Trondheim, Norway, June 26-30, 1995, vol. 2, pp. 637-640.[5] L. Savioja, A. Järvinen, K. Melkas, and K. Saarinen,“Determination of the low-frequency behaviour of anIEC listening room,” In Proc. Nordic AcousticalMeeting (NAM’96), Helsinki, Finland, June 12-14,1996, pp 55-58.[6] H. Samet, The Design and Analysis of Spatial DataStructures, Addison-Wesley, 1990.[7] S. Van Duyne and J. Smith, “The tetrahedral digitalwaveguide mesh”, In Proc. 1995 IEEE ASSP Workshopon Applications of Signal Proc. to Audio andAcoustics, New Paltz, NY, Oct. 1995.[8] S. Van Duyne and J. Smith, “The 3D tetrahedral digitalwaveguide mesh with musical applications”, InProc. 1996 Int. Computer Music Conf., Hong Kong,Aug. 19-24, 1996, pp. 9-16.[9] L. Savioja and V. Välimäki, “The bilinearly deinterpolatedwaveguide mesh,” In Proc. NORSIG’96 IEEENordic Signal Processing Symposium, Espoo, Finland,Sept. 24-27, 1996.