The aim is to provide a better mathematical framework and alternative design methodologies for coarse quantization of signals in a highly oversampled setting. The central themes is "sigma-delta quantization" in analog-to-digital conversion of audio signals and its counterpart "error-diffusion" in 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. A wide range of different schemes exist for this purpose, each corresponding to a different implementation and approximation order. As our primary mathematical objective, we expect to better understand the law of error decay of the existing schemes as well as to improve on these by new designs. On the algorithmic side, we aim to extend this analysis to settings with computational and other implementational constraints.
The fact that digital signals and data sets can be processed, stored and retrieved with great precision and speed places high demands of accuracy on the conversion process from and to the analog world. However, the devices used in the translation process (such as in the case of analog-to-digital and digital-to-analog conversion circuits in audio applications, and printers in image reproduction) are of necessity analog devices, which have physical limitations that, at first sight, conflict with those accuracy demands. To cope with this problem, engineers have empirically developed special signal processing techniques leading to alternative signal and number representations that are quite different from standard decimal or binary representations. Typical techniques take advantage of the highly accurate performance of the analog devices in sampling very densely in time or space to compensate for the lack of amplitude precision of those devices. Interestingly, while these empirical schemes have proved efficient and have been used in consumer products for a long time, the corresponding framework of signal processing has little mathematical support. We aim to study these schemes in more detail, with the goals of improving the mathematical theory as well as proposing variants that outperform those used presently, and to identify a wider range of applications.
