Principles of Digital Waveguide Models of Musical Instruments

Abstract

Basic principles of digital waveguide modeling of musical instruments are presented in a tutorial introduction intended for graduate students in electrical engineering with a solid background in signal processing and acoustics. The vibrating string is taken as the principal illustrative example, but the formulation is unified with that for acoustic tubes. Modeling lossy stiff strings using delay lines and relatively low-order digital filters is described. Various choices of wave variables are discussed, including velocity waves, force waves, and root-power waves. Signal scattering at an impedance discontinuity is derived for an arbitrary number of waveguides intersecting at a junction. Various computational forms are discussed, including the Kelly-Lochbaum, one-multiply, and normalized scattering junctions. A relatively new three-multiply normalized scattering junction is derived using a two-multiply transformer to normalize a one-multiply scattering junction. Conditions for strict passivity of the model are discussed. Use of commutativity of linear, time-invariant elements to greatly reduce computational cost is described. Applications are summarized, and models of the clarinet and bowed-string are described in some detail. The reed-bore and bow-string interactions are modeled as nonlinear scattering junctions attached to the bore/string acoustic waveguide.

Vocal production mechanisms in a non-human primate: morphological data and a model

“…A popular vocal tract model used for simulation of vowel sounds is the ''Wave-Reflection Analogue'' (or ''Wave Digital Filter'') (Smith, 1992(Smith, , 1998. The model divides the three-dimensional vocal tract into a finite number of cylinders of certain length.…”

Fractionally addressed delay lines

“…Especially for applications such as waveguide modeling of musical instruments [19], it is important to consider the attenuation that different frequencies are subject to when fed into the delay line. For instance, the decay time of a frequency partial in a waveguide string depends on the attenuation of the interpolated delay line at that frequency.…”

Plucked-string synthesis algorithms with tension modulation nonlinearity

“…Our approach to discrete-time simulation of tension modulation is to extend the bidirectional linear waveguide model [9], [3] with a signal-dependent nonlinear TVFD structure discussed in [11]. The two delay lines are implemented using two nonlinear delay lines of Fig.…”

“…In physical modeling of plucked and struck string instruments, the vibration of a string is typically simulated with a linear system, such as a digital waveguide model [9], [3]. However, a vibrating string is linear only to a first approximation, and consequently, nonlinear phenomena exhibited by every real string, such as the modulation of string tension, are inherently omitted in these simulations.…”