# The WaveVerb Multi-Channel Room Acoustics Modelling System

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Page 00000472 The WaveVerb Multi-Channel Room Acoustics Modelling System D.T. Murphy and D.M. Howard Department of Electronics, University of York, Heslington, York, YO10 5DD, United Kingdom Email: dtml03@ohm.vork.ac.uk Abstract: Reverberation is perhaps the most important post production/synthesis tool available to the sound engineer or computer musician. This paper introduces the WaveVerb digital waveguide mesh reverberation system which models sound propagation through an enclosed 2-D space using a mesh based on a triangular decomposition of the 2-D plane. Excitation of the mesh with an appropriate input results in an impulse response for the modelled room. Directionally encoded Ambisonic B-format Room Impulse Responses (RIRs) for multichannel surround sound processing can also be generated. WaveVerb's Decode Module allows the user to process an audio file using a modelled RIR via convolution for a variety of different playback formats including 2-D Ambisonic surround sound and binaural headphone monitoring. 1. Introduction The simulation of the acoustics of a room or hall with particular emphasis on its reverberant characteristics is a fundamental tool in the field of creative audio processing. Every sound we hear has associated with it an environmental context and a sense of location for the acoustic space within which it is heard. For hundreds of years composers have manipulated and used these properties of sounds in space as a fundamental part of their music. However, it is only in the relatively recent electronic age that composers and in particular computer musicians have sought complete control over this particular musical element. The acoustic properties of a real enclosed space can be uniquely defined by measuring the Room Impulse Response (RIR) at a specific listening point for an input signal applied at a given sound source location. Two methods traditionally used for modelling a RIR based on a description of the room geometry, are ray tracing [1] and the image source method [2]. These methods have individual limitations although common to both is the fact that they are only valid for high frequencies. At low frequencies, or for small enclosed spaces, where the wave-like behaviour of sound and the effects of room resonances or modes are more noticeable these methods are less appropriate. This problem has been addressed in a number of ways, usually based around a direct time domain model of wave motion within the space. These methods are often computationally intensive and expensive to implement. Digital waveguide models have been used successfully as a partial alternative to time domain models based on the wave equation in the fields of musical acoustics and sound synthesis. 2-D and 3-D rectilinear topologies of unit delay waveguides have been used to model plates [3], membranes [4], acoustic spaces [5], and more abstract sound objects [6]. Current waveguide models of room acoustics are limited to providing only a partial solution to an accurate RIR, as they are less effective at high frequencies where execution time becomes prohibitive for a full 3-D implementation. This paper describes the ongoing implementation of the WaveVerb Multi-channel Spatial Simulation System which uses a digital waveguide model to generate multi-channel RIRs valid for both low and high frequencies. This will be accomplished by limiting the model to the horizontal plane only. 2. Digital Waveguides A waveguide is any medium in which wave motion can be characterised by the one dimensional wave equation. In the lossless case, all solutions can be expressed in terms of left-going and right-going travelling waves and can be simulated using a simple bi-directional digital delay line. A digital waveguide model is obtained by sampling, both in space and time, the one-directional travelling waves which occur in a system of ideal lossless waveguides. The sampling points are called scattering junctions, and are connected by bidirectional unit-delay digital waveguides. Figure 1 shows the general case of a scattering junction J with N neighbours, i = 1,2,....N. The sound pressure in a waveguide is represented by p,, the volume velocity by v, and the impedance of the waveguide by Z,. The input to a waveguide is termed pi and the output pi. The signal pi.j therefore represents the incoming signal to junction i along the waveguide from the opposite junction J. Similarly, the signal pj. represents the outgoing signal from junction i along the waveguide to the opposite junction J. The volume velocity v, is equal to pressure, pi, divided by impedance, Z,. The delay elements are bidirectional and so the total sound pressure in the waveguide is defined as being the sum of its input and output: Pi = P +PI- (1) - 472 - ICMC Proceedings 1999

Page 00000473 Figure 1: Scattering junction J with N neighbours connected via bi-directional unit delay elements. At a lossless scattering junction with N connected waveguides the following conditions must hold: N N Svij = v ~ (Flows add to zero) i-I i-- P1 = ="= Pi- =*= PN (Continuity of impedances) Using these conditions the sound pressure at a scattering junction can be expressed as: pz = 2,(2) "^It/Si Zi As the waveguides are equivalent to bi-directional unit-delay lines, the input to a scattering junction is equal to the output from a neighbouring junction into the connecting waveguide at the previous time step: P+, =z-'P (3) Connecting together unit waveguide elements via scattering junctions allows various objects to be constructed. Waveglide elements connected end on end using two port scattering junctions creates an object that can be used to model the behaviour of a string. A number of parallel string models connected together by replacing the 2 -port scattering junctions for 4-port scattering junctions, allows 2-D mesh structures to be constructed. 3. Digital Waveguide Meshes and Dispersion Error The easiest way of creating a waveguide mesh is using 4 port scattering junctions so that the resulting object is based around a decomposition of the 2-D plane into equal squares. There are however associated problems with using such mesh type structures, effectively limiting their ability to propagate high frequencies. The higher the frequency that is required to be supported by the structure, the denser the mesh has to be in terms of the number of scattering junctions used. This high frequency response is further reduced due to dispersion error. The sampling rate of the mesh, which is determined by the spacing of the scattering junctions and the mesh topology, is given by fupda, = cvd/d, where c is the speed of sound, and d is the distance between scattering junctions. The effective sampling rate, and so the wave of highest frequency that can be propagated successfully by the mesh, is well below the value given by fu,,,. An inherent problem with lattice type structures such as the digital waveguide mesh is that they exhibit angular dependent dispersion. In the ideal case all wave frequencies travel at the same speed in every direction. This property has been examined in detail and is dependent upon the mesh topology [3). Further, it has been shown that the basic rectilinear mesh can be improved upon by using a triangular based mesh consisting of 6-port scattering junctions. This topology is the most efficient decomposition of the 2-D plane in terms of the consistency and minimisation of the dispersion error [4]. As yet it has only been used for modelling 2-D membranes and not for room acoustics problems. 4. The 2-D Triangular WaveVerb Mesh The general junction in the triangular waveguide mesh is the six port scattering junction with N=6 according to equations (1), (2) and (3). For a general junction point with 6 neighbours, the equations defining 2-D wave propagation along the mesh are: Junction Pressure: Pj,= 9p +PJ2. +JP. +P.4 +P. +PJ. ) (4) Junction Outputs: pj.; = pj- p,. (5) Pass signal to next junction: p, / = Z' pj,,- (6) ICMC Proceedings 1999 - 473 -

Page 00000474 Figure 2 shows how the triangular waveguide mesh is constructed and how each general scattering junction is connected to each of its six neighbours. (a) (b) Figure 2: The triangular waveguide mesh: (a) Triangular structure; (b) 6 port scattering junction. 5. Wave Verb - The Wave Propagation Interface The WaveVerb System is used to model wave propagation through a 2-D representation of a room. Figure 3(a) shows the user interface employed which enables the user to observe wave propagation on a graphical display with a number of different views and realisations. This provides excellent visual feedback on the behaviour of the propagating wave and the properties of the room itself. This module forms the basis of the WaveVerb system and allows the user to make RIR measurements for various source-listener positions. Control is given over variables such as room width, length and mesh density. In this way the user does not require any in depth knowledge of room acoustics in order to generate a RIR for a reverberant effect - the parameters the user has control over correlate readily to real world variables such as room size, source location and object position. Wave phenomena such as diffraction, reflections and interference are natural consequences of the model. Figure 3(b) shows a close up of a room where diffraction effects are clearly evident due to the gaps in the dividing wall. Ie~dl I -sl~sl3r~s (b) (a) (c) Figure 3: The WaveVerb System: (a) The Wave Propagation Interface; (b) Close up of wave propagation in a rectangular room - note the diffraction effects due to the gaps in the dividing wall. (c) The Decode Module. 6. The Analysis Module Using the Analysis Module it is possible to examine RIRs in more detail in both the time and frequency domains. Figure 4 shows a measured RIR with its corresponding frequency response up to the approximate critical frequency of the room. Below this frequency value modal behaviour is said to be the dominant acoustic factor. The modelled room is 6.Om long, 4.4m wide and an input point is defined as being 0.99m from the rear wall and 0.99m from the left wall. An output point is defined as being 4.99m from the rear wall and 3.41m from the left wall. A mesh sampling rate of d = 0.022m is used, givingf,,, = 22049Hz. Walls are set to allow phase preserving reflections, and two 2.0s RIRs are generated. The first RIR has all walls set to be slightly absorptive and this is shown in Figure 4(a). The second has all walls set to be perfectly reflecting with no absorption and it is this low frequency response that is shown in Figure 4(b). For comparison, the analytical room modes are calculated and plotted on the same graph. Note that only axial (reflections between two surfaces denoted by the shorter dotted lines) and tangential modes (reflections between four surfaces denoted by the longer dashed lines) - 474 - ICMC Proceedings 1999

Page 00000475 are valid. It can be seen that there is an exact correlation between the analytical room modes as calculated and the resonant frequencies highlighted by examining the frequency response of the measured RIR. -I I~ I I, I.I I t II t I I (a) (b I Figure 4: RIR results: (a) RIR generated by triangular waveguide mesh with all walls set to be slightly absorbing; (b) Frequency response of a room with no absorption below approximate critical frequency. Analytical axial modes:.............. Analytical tangential modes: - - - - 7. The Decode Module It is possible to encode a soundfield by decomposing it into spherical harmonic components. The zero order pressure component is termed W and is omnidirectional, picking up all sounds from all directions equally. The first order velocity components are figure-of-eight responses pointing forward, left and up, and are termed X, Y and Z respectively. The four signals, W, X, Y and Z are known as B-format. Reproducing a B-format signal is possible using an appropriate Ambisonic decoding scheme and a multi-speaker array [7]. It is possible to derive horizontal only B-format signals - W, X and Y only - from the mesh at any point resulting in a B-format multichannel RIR. Using this encoding method the soundfield in the modelled room can be recreated complete with acoustic cues associated with the location of a sound source in relation to the listener. This is in addition to the cues relating to source distance and environmental context that are already present in a single channel RIR. Using the decoding module as shown in Figure 3(c), anechoic or synthesized sound can be processed with a Bformat RIR for playback over a multi-speaker array. Further processing using a set of measured Head Related Transfer Functions (HRTFs) allows binaural surround sound monitoring over headphones removing the necessity for a large multi-speaker array. 8. Conclusions This paper has introduced the WaveVerb Multi-channel Spatial Simulation System which uses a digital waveguide mesh based upon a triangular decomposition of the 2-D plane as a method of modelling wave propagation in an enclosed space. The triangular mesh is valid for modelling the low frequency behaviour of a room as already established for rectilinear meshes and exhibits wave phenomena such as diffraction and interference. It has already been shown that the triangular mesh offers significant improvement over a similar model using a rectilinear topology [8]. Further work involves acoustic analysis of the modelled RIRs. together with subjective testing using anechoic audio convolved with mono, stereo and Ambisonic B-format RIRs. References [1] A. Krokstad, S. Stram and S. Sorsdal, "Calculating the Acoustical Room Response by use of a Ray Tracing Technique", Journal of Sound Vibration, Vol. 8, No. 1, pp 118-125. 1968. [2] U. Stephenson, "Comparison of the Mirror Image Source Method and the Sound Particle Simulation Method", Applied Acoustics, Vol. 29, pp 35-72, 1990. [3] S.A. Van Duyne and J.O. Smith III, "The 3D Tetrahedral Digital Waveguide Mesh with Musical Applications", Proc. ICMC 1996, pp 9-16 (1996). [4] J. Laird, P. Masri and C. N. Canagarajah, "Efficient and Accurate Synthesis of Circular Membranes Using Digital Waveguides", Proceedings of the IEE Colloquium on Audio and music technology: The creative challenge of DSP, IEE Digest 98/470, 12/1-12/6. [5] L. Savioja, M. Karjalainen and T. Takala, "DSP Formulation of a Finite Difference Method for Room Acoustics Simulation", Proc. NORSIG'96 IEEE Nordic Signal Processing Symposium. Espoo, Finland, Sept 1996. [6] D. Rossiter, A. Horner and G. Baciu, "Visualization and Manipulation of 3D Digital Waveguide Structures for Sound Experimentation", Proc. ICMC 1996, pp 43-46 (1996). [7] M.A. Gerzon, "General Metatheory of Auditory Localisation". Presented at the 92nd Audio Engineering Society Convention, March 1992, Preprint 3306, 1992. [8] D.T. Murphy & D.M. Howard, "Digital Waveguide Modelling of Room Acoustics: Comparing Mesh Topologies", Proc. 25"' Euromicro Conference, Milan, Italy, 8"'-10th Sept. 1999. ICMC Proceedings 1999 -475 -