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Digital waveguide synthesis

Digital waveguide synthesis is the synthesis of audio using a digital waveguide. Digital waveguides are efficient computational models for physical media through which acoustic waves propagate. For this reason, digital waveguides constitute a major part of most modern physical modeling synthesizers.

A lossless digital waveguide realizes the discrete form of d'Alembert's solution of the one-dimensional wave equation as the superposition of a right-going and a left-going waves,

: $y(m,\; n)\; =\; y^+(m\; -\; n)\; +\; y^-(m\; +\; n),$

where $y^+$ is the right-going wave, and $y^-$ is the left-going wave. It can be seen from this representation that sampling the function $y$ at a given position $m$ and time $n$ merely involves summing two delayed copies of its traveling waves. These traveling waves will reflect at boundaries such as the suspension points of vibrating strings or the open or closed ends of tubes. Hence the waves travel along closed loops.

Digital waveguide models therefore comprise digital delay lines to represent the geometry of the waveguide which are closed by recursion, digital filters to represent the frequency-dependent losses and mild dispersion in the medium, and often non-linear elements. Losses incurred throughout the medium are generally consolidated so that they can be calculated once at the termination of a delay line, rather than many times throughout.

Waveguides such as acoustic tubes are three-dimensional, but because their lengths are often much greater than their diameters, it is reasonable and computationally efficient to model them as one-dimensional waveguides. Membranes, as used in drums, may be modeled using two-dimensional waveguide meshes, and reverberation in three-dimensional spaces may be modeled using three-dimensional meshes. Vibraphone bars, bells, singing bowls and other sounding solids (also called idiophones) can be modeled by a related method called banded waveguides, where multiple band-limited digital waveguide elements are used to model the strongly dispersive behavior of waves in solids.

The term "digital waveguide synthesis" was coined by Julius O. Smith III , who helped develop it and eventually filed the patent. A digital waveguide model of an ideal vibrating string having a single point of damping implemented as a two-point average and initialized to random initial positions and velocities at every sample can be shown to be equivalent to the Karplus–Strong algorithm which was developed some years earlier. Stanford University owned the patent rights for digital waveguide synthesis and signed an agreement in 1989 with Yamaha to develop the technology. All early patents have expired and new products based on the technology are appearing frequently.

An extension to DWG synthesis of strings made by Smith is commuted synthesis, wherein the excitation to the digital waveguide contains both string excitation and the body response of the instrument. This is possible under the assumption that the string and body are linear time-invariant systems, which is approximately true for typical instruments, allowing the excited body to drive the string, instead of the excited string driving the body as usual. Thus, the string is excited by a "plucked body response". This means it is unnecessary to model the instrument body's resonances explicitly using hundreds of digital filter sections, thereby greatly reducing the number of computations required for a convincing resynthesis.

Prototype waveguide software implementations were done by students of Smith in the Synthesis Toolkit (STK).

The first musical use of the Extended Karplus Strong (EKS) algorithm was in the composition "May All Your Children Be Acrobats" (1981) by David A. Jaffe, followed by his "Silicon Valley Breakdown" (1982). Since the EKS became understood as a special case of digital waveguide synthesis years later, the piece can now be considered the earliest use of digital waveguide synthesis as well.

Related was "A Bicycle Built for Two" by Max Mathews, John Kelly, and Carol Lochbaum at Bell Labs in 1961, which used the Kelly–Lochbaum ladder filter to model the human vocal tract. To distinguish ladder filters from digital waveguide filters, a digital waveguide is defined as a bidirectional delay line at least two samples long over which no scattering occurs. Provided by Wikipedia