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

posite end one time step before. This can be expressed asthe equation+p iz – 1 -= p i,opp(3)-where p i,opppresents output at opposite end of thewaveguide i. By using equations (1) and (3) the soundpressure in a node takes the following formNz – 1Np i,oppz – 1 +p i, opp2 ∑ ------------------------ 2Z∑ ------------------------iZpi = 1i= --------------------------------------- – ---------------------------------------i = 1(4)NN11∑ ----Z∑ ----iZi = 1i = 1 iBy applying once again the same equations we get an expressionfor sound pressureN p iopp ,2 ∑ ---------------p z – 1 Zi = 1 i= ------------------------------ – z – 2 p(5)N1∑ ----Zi = 1 iEquation (5) can be expressed also in form of a transferfunctionz – 1 N p Ni, opp 1⋅ 2 ∑ --------------- ⁄Z∑ ----iZpi = 1i= ----------------------------------------------------------------i = 1(6)1 + z – 2In a usual case the medium is homogenous and impedancesin all waveguides are equal and equation (6) can bereduced to the formz – 1 2 N⋅ --- N∑ p i,oppp = ---------------------------------------------i = 1(7)1 + z – 2Verbal description of equation (7) is that the sound pressurein a node is two times the average of its neighborssubtracted by its own value two time units ago. This expressionis independent of the dimensionality of the system.Only the number of neighbors is relevant.2.2. Mesh ExcitationExciting the mesh must be done so that none of the equations(1)-(5) is violated. If we want to create an excitationof value I in a homogenous mesh at a node with N neighbors,we set the incoming sound pressure to allwaveguides connected to it. Using equation (2) the valuefor sound pressure is found to be+ Ip i= -- ∀i= 1…N(8)2To keep the equation (3) satisfied the history of neighborshas to be updated also, since the outgoing sound pressuresin opposite ends of waveguides at one time stepbefore excitation must equal to I/2. Because there are noother outputs, the sum of output signals is also I/2, which1D3D2DFig 2. Example structures of rectilinear 1D, 2D and 3Dmeshes.is the same as the sum of input signals. By using equation(2) again we come to the result that all the neighbor nodesmust have value I/N at one time step before excitation.3. MESH STRUCTUREThere are different topologies which may be used to constructa digital waveguide mesh. The only restriction isthat all the delay-elements connecting nodes must be ofequal length. If there are waveguides of different lengths,the equation (5) is no more valid. Instead, equation (4) canbe used, z – 1– xis replaced with z i, and x represents the delaylength of waveguide i.The requirement of equal length waveguides is satisfiedby any regular space decomposition method. It meansthat the atomic tiles of decomposition are polytopeswhose sides are of equal length [6]. In this study we concentrateon rectilinear meshes, since they are computationallyefficient and easy to construct. Some hexagonal,triangular and tetrahedral structures both in 2- and 3-dimensionalsystems are presented in [7] and [8].An N-dimensional rectilinear waveguide mesh is aregular array of digital 1-D waveguides arranged alongeach perpendicular dimension, interconnected at theircrossings. In figure 2 there are examples of meshes of differentdimensionalities.In an N-dimensional rectilinear mesh each node insidethe mesh has exactly 2*N neighbors. Nodes on the boundarieshave only one neighbor. Due to this arrangement theboundaries can be terminated the same way as in digital1D waveguides.3.1. Sampling Frequency of a Rectilinear MeshDistance between mesh nodes together with the mesh topologydetermines the sampling rate of the mesh. In a onedimensionalsystem waves propagate one unit distance inone time unit since the waveguide mesh is constructed ofunit-delays.In a multi-dimensional rectilinear mesh effective wavepropagation speed is found at the diagonal direction. In anN-dimensional system one diagonal movement, correspondingto distance of N unit distances, requires N unitdelays. Thus the update frequency of a rectilinear N-dimensionalmesh is