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Scattering Delay Networks

A recent development in artificial reverberation technology is the Scattering Delay Network (SDN) [,237]. An SDN can be considered a computationally efficient approximation to geometric ray tracing using a small digital-waveguide network. It thus classifies as both a delay network and approximate physical room model.

Figure: The Image Method for determining acoustic reflection paths (``ray tracing''). The source and listener are labeled $ S$ and $ L$ , respectively, with left-right reversals where appropriate to indicate their mirrored orientation. Only reflections due to the left and right walls are considered. The solid lines not indicating walls show the physical reflection paths. The dashed lines show the contributions of the first-order images to the first-order reflections $ f_1$ and $ f_2$ . The dotted lines show contributions of the second-order images to the second-order reflections $ s_1$ and $ s_2$ . The direct path is labeled $ d$ .
\includegraphics[width=\twidth]{eps/ImageMethod}

Figure [*] illustrates the image method [11,] for determining acoustic reflections between two parallel walls. The method can be visualized by imagining the walls of the acoustic space to be mirrored. Then the acoustic ray paths from the source $ S$ to the listener $ L$ can be found by drawing a straight line from each source image to the listener. The length of this line determines propagation delay and filtering due to air absorption, and the wall traversals going from room-image to room-image indicate the actual reflection planes, where further absorption and dispersion can be applied. In Fig.[*], only the left and right wall reflections are considered, yielding a simplified series of room images along one dimension.

Figure: Scattering Delay Network (SDN) for modeling two parallel walls. Solid lines other than the walls denote four bidirectional digital waveguides intersecting at two scattering junctions denoted by $ A$ in a semicircle []. All reflection paths are composed of these path segments. Other notation is as in Fig.[*].
\includegraphics[width=\twidth]{eps/ImageMethodSDN}

Figure [*] shows a Scattering Delay Network (SDN) model for the configuration of Fig.[*]. In an SDN, the second-order (and all higher-order) image sources are constrained to follow the same path segments as the first-order reflections. The direct path $ d$ and first-order reflections $ f_1$ and $ f_2$ are modeled precisely (correct propagation delays and filtering--filtering not shown) because the locations of the two waveguide junctions along the walls are chosen to yield correct first-order reflections. The second-order reflections $ s_1$ and $ s_2$ are constrained to use the same waveguides, yielding the modified path images shown in dotted lines. The path of each second-order reflection is altered as shown, forcing it to reuse acoustic path segments of the first-order reflections while introducing a new path segment which connects the two scattering junctions. Only the second-order reflections make use of the waveguide bidirectionality in this simple example. We can see, for example, that the modified paths are always somewhat longer than the straight lines-of-sight from source-image to listener $ L$ . In particular, a ray emitted from a source-image aims for the listener-image in the adjacent room image (correct for a first-order reflection) instead of directly for the real listening-point L. Higher order reflections are thus lengthened by a non-diverging factor, and this lengthening disappears as the source and listener approach any line orthogonal to the walls.

Since only the left-right wall reflections are considered in Fig.[*], it can be considered a 1D example (virtual room images are laid out along only one dimension). The 2D case would introduce two more scattering junctions and support image sources from all directions in the 2D plane. The 3D case requires six scattering junctions and $ 2N-1$ digital waveguides. There is no reason not to consider dimensions higher than $ N=3$ .

The filtering and arrival times in an SDN model are therefore well preserved qualitatively, and the SDN can achieve a high degree of perceptual accuracy for its computational cost. However, since the details of the reflection pattern are modified after the first-order reflections, the early texture of the impulse response should be checked for "flutter" and any other textural defects. As discussed in [], perceptually adequate mode densities are achieved for cuboid reverberant spaces larger than on the order of ten cubic meters.

Figure [*] depicts a complete 2D Scattering Delay Network (SDN), adding now the reflections from the other two walls. Note that the scattering junctions are fully connected. This is necessary to support image sources along any diagonal. (A pair of junctions on opposite walls only support room images along the axis orthogonal to those walls.) The diagonal waveguides are needed when the best path from a source image needs to enter a room image from the north or south, say, but exit to the east or west.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2017-05-16 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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