Dispersive Systems in Musical Audio Signal Processing

Abstract

Dispersion is a property seen throughout both nature and the man-made world. By its most basic definition, it simply refers to the spreading out or scattering of some form of wave phenomenon in a medium. In practice, this is usually due to variation in the propagation speed of the wave with respect to frequency or amplitude. Examples of dispersion in the natural world are rainbows, the spreading of water surface waves, and the distinctive sound generated when striking a long metal wire. This thesis explores the presence of dispersion in signal processing systems designed for musical use, and develops a number of new musical digital signal processing systems which are based around the action of dispersion. The primary focus of the thesis is spring reverberation, an early form of artificial reverberation based on exciting vibrations in helical metal springs. Springs are unique in the world of analog musical technology, in that they derive almost all of their recognized sonic character and desirability from the dispersion they induce. The first portion of this thesis examines the behavior of spring reverberators through the lens of mathematical models of spring vibration, and develops some important new results about their behavior. These mathematical models are turned directly into a digital model, via the application of finite difference methods. A similar system, that of the larger 'Slinky' spring, is examined and modeled via the same framework. The second part of this thesis examines an alternative method for emulating spring reverberation, based heavily on the use of the dispersive allpass filter. The result is an efficient and high-quality parametric emulation of the spring reverberator. This model is further extended in several novel ways to improve its efficiency, including via application of multi-rate techniques. The reverberator is further improved by the proposal of a new structure that uses sparse-noise convolution to generate diffusion in the repeating echoes. Finally, a new algorithm is developed that uses the dispersion of golden-ratio allpass filters as a method for transparently reducing the peak amplitude of musical signals so that their loudness can be maximized. The novel algorithms developed in this thesis are intended for use in real-time music production or processing applications. Hence, computational efficiency and parametric control are central considerations. These algorithms could be implemented for use in a computer environment, for mobile devices, or for embedded DSP systems.