Delta-Sigma Waveguides for Music Synthesis

Abstract

33 Computer Music Journal, 23:4, pp. 33–47, Winter 1999 © 1999 Massachusetts Institute of Technology. Physical modeling of musical instruments has proven to be an interesting and useful technique for sound synthesis, because it provides the variability of real instruments. Natural dynamics can be captured in the model, yielding realistic emulation of traditional instruments. Model parameters can correspond to actual physical parameters so that parameter adjustment is intuitive, and these parameters can be adjusted to produce exotic new sounds. The digital waveguide (Smith 1992) is a computationally efficient method for the physical modeling of strings, drum heads, bores, and other resonant structures that appear in musical instruments. These waveguides can be implemented as a program running on a general-purpose computer which generates and stores sounds for later playback. Computers with sufficient capabilities can generate some of these sounds in real time for playing the sounds as they are produced. Alternately, special-purpose computer hardware can implement these waveguides to produce the sounds in real time. Recent work has introduced the possibility of digital hardware that can be reconfigured, and this alternative implementation may be particularly attractive for music synthesis. The reconfigurable hardware offers operating performance that approaches that of specialized, dedicated devices. However, the hardware can also be rearranged to implement different processing structures. The time required to rearrange the hardware is typically long with respect to the processing time while in any particular configuration, but this constraint appears to be consistent with typical music synthesis systems; that is, flexibility of synthesis is desired, but some delays can be tolerated to “patch” new sounds. The reconfigurable hardware does impose certain implementation constraints when compared with software emulations running on a generalpurpose computer. These constraints are similar to those associated with any hardware implementation of physical models. This article addresses the constraints and shows accommodations that support the use of reconfigurable devices to implement digital waveguides. A direct pulse-code modulated (PCM) implementation of a digital waveguide in hardware leads to underutilization of hardware resources in two significant ways. First, the hardware data paths are clocked at rates near the audio rate of 44.1 kHz, which is significantly below the maximum clock rate typical for modern digital circuits. Second, the wide PCM data paths require large coefficient multipliers in the filter. A delta-sigma approach can be used to produce a more efficient implementation, with the recirculated data reduced to a high-speed stream of single bits. This filter hardware is significantly smaller, because multiplication of a coefficient by a single bit in the stream can be implemented as a multiplexer.