Digital signal processing has been one of the technological revolutions of highest impact. At the core, this technology relies fundamentally on approximating analog signals of the real world by discrete representations that can be stored, transmitted and processed without loss of information by digital circuits and computers. Despite the ubiquity of applications and products that process signals digitally, the interface between the analog and the digital worlds, i.e. the quantization paradigm, is still only partially understood from a fundamental point of view. While the classical approach of high-resolution discretization of independent signal samples has remained dominant in the theory, the actual technology has evolved in a completely different direction in which highly correlated signal samples are quantized very coarsely using as few as 1 bit per sample. Everyday technological products such as CD players, cell phones and laser printers have employed this alternative approach very successfully for over a decade already, yet the current theoretical understanding of coarsely quantized overcomplete expansions is still highly underdeveloped. The goal of this project is to set forth the theoretical foundations of this subject with the motivations of better analyzing existing methods, introducing new designs and creating new potential applications by extending the concept to other areas of signal processing.
Two specific and related methods for coarse quantization of redundant (overcomplete) signal expansions are sigma-delta modulation for A/D conversion of audio signals and error-diffusion for digital halftoning of images. In both cases, a given target analog signal is represented by a judiciously chosen one-bit (or few-bit) sequence which approximates the signal in a suitable low-pass subspace. Both methods employ specially designed nonlinear feedback dynamical systems to generate these representations, exact analyses of which are very challenging. Central questions of input-output error signal analysis -- a somewhat alien subject to dynamical systems -- have been traditionally addressed via empirical linear modeling. This research introduces radically new tools of analysis and design by incorporating dynamical systems and signal processing methods through a genuine study of the nonlinearity. A secondary contribution is the development of new signal processing applications in communications theory inspired by coarse quantization ideas in redundant systems.
