ITR: Collaborative Research: Accurate Representations of Signals in a Coarse-Grained Environment

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The aim is to provide a better mathematical framework andalternative design methodologies for coarse quantization of signals ina highly oversampled setting. The central themes is "sigma-deltaquantization" in analog-to-digital conversion of audio signals and itscounterpart "error-diffusion" in digital halftoning of images. In bothcases, a given target analog signal is represented by a judiciouslychosen one-bit (or few-bit) sequence which approximates the signal in asuitable low-pass subspace. A wide range of different schemes exist forthis purpose, each corresponding to a different implementation andapproximation order. As our primary mathematical objective, we expectto better understand the law of error decay of the existing schemes aswell as to improve on these by new designs. On the algorithmic side, weaim to extend this analysis to settings with computational and otherimplementational constraints.The fact that digital signals and data sets can be processed, storedand retrieved with great precision and speed places high demands ofaccuracy on the conversion process from and to the analog world.However, the devices used in the translation process (such as in thecase of analog-to-digital and digital-to-analog conversion circuits inaudio applications, and printers in image reproduction) are ofnecessity analog devices, which have physical limitations that, atfirst sight, conflict with those accuracy demands. To cope with thisproblem, engineers have empirically developed special signal processingtechniques leading to alternative signal and number representationsthat are quite different from standard decimal or binaryrepresentations. Typical techniques take advantage of the highlyaccurate performance of the analog devices in sampling very densely intime or space to compensate for the lack of amplitude precision ofthose devices. Interestingly, while these empirical schemes have provedefficient and have been used in consumer products for a long time, thecorresponding framework of signal processing has little mathematicalsupport. We aim to study these schemes in more detail, with thegoals of improving the mathematical theory as well as proposingvariants that outperform those used presently, and to identify a widerrange of applications.