Investigations in digital signal processing and electroacoustics

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This research falls into three distinct areas:(a) Many modern analogue-to-digital and digital-to-analogue conversion systems use an architecture known as a sigma-delta modulator.  These nonlinear dynamical systems, which are closely related to noise shapers, display behaviour that can appear to be chaotic, but which is actually completely deterministic.  Nevertheless, only the 1st-order sigma-delta modulator has thus far proved amenable to detailed mathematical analysis.  It is believed that a new insight into their behaviour, resulting from a recent discovery known as the "integer noise-shaping theorem," will allow a mathematical understanding to be achieved for a larger class of such systems.(b) The mathematical theory of dithered quantization and noise shaping, which we have developed primarily for digital audio applications over the past 20 years, is also applicable to image and video processing.  The full extension of the 1-dimensional audio results to 2-dimensional image processing requires the development of new theorems in the theory of functions of two complex variables -- specifically, the extension of Jensen's theorem, and of the properties of minimum-phase noise shapers, to two dimensions.  This will be the focus of this aspect of our research.(c) Low-frequency acoustic transducers have diaphragms that undergo excursions large enough that the linearized acoustic wave equation is not sufficiently precise to predict their behaviour.  In particular, a high-frequency transducer in close proximity to such a low-frequency transducer suffers both amplitude- and frequency-modulation effects due to this nonlinearity.  We shall investigate improved models to predict these effects and to control them.