Nguyen T. Thao
Data acquisition (analog-to-digital and digital-to-analog conversions) is an intrinsic part of signal processing, as the front and terminal ends of any DSP chain. Paradoxically, the technology of data conversion has evolved towards the use of circuits, such as Sigma-Delta modulators, that have more or less escaped from a clear understanding by the signal processing community from a rigorous system theoretic point of view, although they have been omnipresent for already several decades. The presence of a nonlinear operation (namely, quantization) in a feedback loop makes the analysis difficult, and explains the currently missing signal theory for these systems. The specific difficulty is the formalization of input-output transfer function for these nonlinear systems. Analog circuit engineers have developed analytical techniques of their own, which however do not fit in the traditionally rigorous signal theory. This indeed has been based on an approximate model of the quantizer error as an external and independent source of noise, while quantization is fundamentally deterministic. This model becomes particularly inaccurate as the quantizer of a Sigma-Delta modulator is typically coarse. Meanwhile, DSP textbooks more and more need to address these modern techniques of data acquisition given their current authority, but are left with a presentation that locally appears to be empirical as opposed to the rest of the DSP material.
This tutorial gives a rigorous and unified view on quantization and data acquisition, catered for the signal processing community, with the introduction of external knowledge from the area of dynamical systems. The fundamental obstacle to a rigorous analysis of quantization errors has been the derivation of the standard error autocorrelations E(e[k]e[k+n]) (where E designates here some averaging operation). Specifically, nonlinearities have prevented the exact account of the deterministic relation between e[k+n] and e[k], before the averaging operation. The dynamical system approach re-establishes this deterministic link, from which point rigorous signal theory can be restored and data conversion systems can be rigorously and consistently re-explained. Elegant functional analysis such as the spectral theory of unitary operators in a Hilbert space can be rigorously and directly applied.
This work was performed in collaboration with Sinan Gunturk, from Courant Institute of Mathematical Sciences, New York.
Nguyen T. Thao received engineering degrees from Ecole Polytechnique, Palaiseau, France, and Ecole Nationale Supérieure des Télécommunication, Paris, France, in 1984 and 1985, respectively, the M.S. degree in electrical engineering from Princeton University, Princeton, NJ, in 1986, and the Ph.D. degree in electrical engineering from Columbia University, New York, in 1993 under the supervision of Martin Vetterli.
From 1986 to 1989, he was with the GaAs integrated circuit group of Thomson-CSF, Orsay, France, where he designed and developed high-speed analog–digital (A/D) converters. From 1989 to 1993, as a doctoral student and Postdoctoral Fellow at Columbia University, his work focused on the study of signal reconstruction in oversampled A/D conversion and Sigma–Delta modulation. In 1993, he became an Assistant Professor in the Department of Electrical and Electronic Engineering, Hong Kong University of Science and Technology, Hong Kong, where he conducted research on quantization both in signal processing and image processing. In 1997, he joined the Imaging Technology Department, Hewlett Packard Laboratories, Palo Alto, CA, where he focused on halftoned image processing. Since 1999, he has been an Associate Professor in the Department of Electrical Engineering, City College, City University of New York. His current research emphasis is the theoretical signal analysis of Sigma-Delta modulation.