Formal Spectral Theory for Feedback Systems with Quantization (Sigma-Delta Modulation)

April 30, 2007
Time: 2:00pm-3:00pm
Interschool lab - 7th floor CEPSR
Hosted by: Columbia Integrated System Laboratory
Speaker: Thao Nguen, City College of New York


A Sigma-Delta modulator is an analog-to-digital converter that includes a scalar quantizer in a feedback loop. This permits the achievement of high-resolution conversions with a coarse and imprecise quantization. The error analysis of this system however escapes from the existing signal and system theories. In the signal processing/communications area, only linear feedback systems are rigorously understood.

We build rigorous foundations to the error analysis of Sigma-Delta modulators by importing knowledge from dynamical systems into the classic linear system framework. We drop the standard signal processing approach, which looks at the quantizer error signal as the transformation of the input signal by some transfer function. Instead, we present this error signal as the output of an input-free and time-invariant dynamical system. This is possible in steady state when the input is a finite sum of sinusoids.

This new signal approach allows the use of "noble" mathematical tools such as functional analysis, for the rigorous analysis of quantization. This is a major contribution, as the discrete nature of quantization has prevented the use of continuous mathematics and has typically required the use of approximate and stochastic models (noise). Under the new framework, we derive rigorously quantization error spectra thanks to the powerful properties of unitary operators in Hilbert Spaces. The famous former work by Robert Gray and related authors performed in the special case of "ideal" modulators finds itself concisely rewritten as one particular case of this new theory.

Thao Nguyen obtained his PhD degree in Electrical Engineering at Columbia University in 1993. He joined the EEE Department of Hong Kong University of Science and Technology as an Assistant Professor from 1993 to 1997. He later became a member of technical staff at HP Laboratories in Palo Alto, CA, from 1998 to 1999. He has been an Associate Professor in the EE Department of The City College of New York since 1999. His research interest mainly focuses on the theoretical analysis of A/D and D/A conversion.

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